Write the first five terms of the sequences whose n^th term is a_n= n (n + 2).

Write the first five terms of the sequences whose n^th term is a_n =

Write the first five terms of the sequences whose n^th term is a_n = 2ⁿ

Write the first five terms of the sequences whose n^th term is a_n = (2n - 3) ÷ 6 

Write the first five terms of the sequences whose n^th term is a_n = (-1) ^n-1 5ⁿ⁺¹

Write the first five terms of the sequences whose n^th term is a_n = n × (n² + 5) ÷ 4

Find the 17^th term in the following sequence whose n^th term is a_n = 4n – 3, a₁₇; a₂₄

Find the 7^th term in the following sequence whose n^th term is a_n = n² ÷ 2n, a₇

Find the 9^th term in the following sequence whose n^th term is a_n = (-1) ^n-1 n³, a₉

Find the 20^th term in the following sequence whose n^th term is a_n = n (n - 2) ÷ (n + 3); a₂₀

Write the first five terms of the following sequence and obtain the corresponding series:

a₁ = 3, a_n = 3a_n-1 + 2 for all n > 1

Write the first five terms of the following sequence and obtain the corresponding series:

a₁ = -1, a_n = a_n-1 ÷ n, n ≥ 2

Write the first five terms of the following sequence and obtain the corresponding series:

a₁ = a₂ = 2, a_n = a_n-1, n > 2

The Fibonacci sequence is defined by 1 = a₁ = a₂ and a_n = a_n-1 + a_n-2;

n > 2 Find a_n+1 ÷ a_n for n = 1, 2, 3, 4, 5

Find the sum of odd integers from 1 to 2001.

Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20^th term is –112.

Find the mean and standard deviation using short-cut method.

xi	60	61	62	63	64	65	66	67	68
fi	2	1	12	29	25	12	10	4	5

 

How many terms of the A.P.  -6, - , -5, ……… are needed to give the sum –25?

In an A.P., if p^th term is  and q^th term is , prove that the sum of first pq terms is (pq + 1) ÷ 2, where p ≠ q.

Find the mean and variance for the following frequency distribution.

Classes	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

Question 8:

Classes	0-10	10-20	20-30	30-40	40-50
Frequencies	5	8	15	16	6

If the sum of a certain number of terms of the A.P. 25, 22, 19, ... is 116. Find the last term.

Find the sum to n terms of the A.P., whose k^th term is 5k + 1.

Find the mean, varience and standard deviation using short-cut method.

Height in cms	70-75	75-80	80-85	85-90	90-95	95-100	100-105	105-110	110-115
No. of children	3	4	7	7	15	9	6	6	3

 

The diameters of circles (in mm) drawn in a design are given below:

Diameter	33-36	37-40	41-44	45-48	49-52
No. of children	15	17	21	22	25

Calculate the standard deviation and mean diameter of the circles.                                                 

[Hint: First make the data continuous by making the classes as 32.5-36.5, 36.5-40.5,

40.5-44.5, 44.5 - 48.5, 48.5 - 52.5 and then proceed.]

If the sum of n terms of an A.P. is (pn + qn²), where p and q are constants, find the common difference.

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18^th terms.

 

From the data given below state which group is more variable, A or B?

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80
Group A	9	17	32	33	40	10	9
Group B	10	20	30	25	43	15	7

 

From the prices of shares X and Y below, find out which is more stable in value:

X	35	54	52	53	56	58	52	50	51	49
Y	108	107	105	105	106	107	104	103	104	101

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that

{a (q - r) ÷ p} + {b (r - p) ÷ q} + {c (p - q) ÷ r} = 0

An analysis of monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:

	Firm A	Firm B
No. of wages earner	586	648
Mean of monthly wages	Rs 5253	Rs 5253
Variance of the distribution of wages	100	121

(1) Which firm A or B pays larger amount as monthly wages?

(2) Which firm, A or B, shows greater variability in individual wages?

The ratio of the sums of m and n terms of an A.P. is m²: n². Show that the ratio of m^th and n^th term is (2m – 1): (2n – 1).

The following is the record of goals scored by team A in a football session:

No. of goal scored	0	1	2	3	4
No. of matches	1	9	7	5	3

For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals.

Find which team may be considered more consistent?

If the sum of n terms of an A.P. is 2n² + 5n and its m^th term is 164, find the value of m.

The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:

Σ_i=1⁵⁰ x_i = 212, Σ_i=1⁵⁰ x_i² = 902.8, Σ_i=1⁵⁰ y_i = 261, Σ_i=1⁵⁰ y_i² = 1457.6 

Which is more varying, the length or weight?

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

The mean and variance of eight observations are 9 and 9.25, respectively.

six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

 

If (aⁿ + bⁿ) ÷ (a^n-1 + b^n-1) is the A.M. between a and b, then find the value of n.

The mean and variance of 7 observations are 8 and 16, respectively.

If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations.

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P.

and the ratio of 7^th and (m – 1) ^th numbers is 5: 9. Find the value of m.

The mean and standard deviation of six observations is 8 and 4, respectively.

If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Given that x  is the mean and σ² is the variance of n observations x₁, x₂ ... x_n.

Prove that the mean and variance of the observations ax₁, ax₂, ax₃ ...ax_n are ax  and a²σ² respectively (a ≠ 0).

A man starts repaying a loan as first installment of Rs. 100.

If he increases the installment by Rs 5 every month, what amount he will pay in the 30^th installment?

The mean and standard deviation of 20 observations is found to be 10 and 2, respectively.

On rechecking, it was found that an observation 8 was incorrect.

Calculate the correct mean and standard deviation in each of the following cases:

(1) If wrong item is omitted.               

(2) If it is replaced by 12.

The difference between any two consecutive interior angles of a polygon is 5°.

If the smallest angle is 120°, find the number of the sides of the polygon.

The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject	Mathematics	Physics	Chemistry
Mean	42	32	40.9
Standard Deviation	12	15	20

Which of the three subjects shows the highest variability in marks and which shows the lowest?

Find the 20th and n^th terms of the G.P. , , , ………...

The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively.

Later on, it was found that three observations were incorrect, which were recorded as 21, 21 and 18.

Find the mean and standard deviation if the incorrect observations are omitted.

Find the 12^th term of a G.P. whose 8^th term is 192 and the common ratio is 2.

The 5^th, 8^th and 11^th terms of a G.P. are p, q and s, respectively. Show that q² = ps.

The 4^th term of a G.P. is square of its second term, and the first term is –3. Determine its 7^th term.

Which term of the following sequences:

(a) 2, 2√2, 4, ... is 128?            

(b) √3, 3, 3√3, ... is 729?   

(c) , , , ... is ?

For what values of x, the numbers , x, -  are in G.P?

Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 ...

Find the sum to n terms in the geometric progression √7, √21, 3√7, ...

Find the sum to n terms in the geometric progression 1, -a, a², -a³, ……. (If a ≠ -1)

Find the sum to n terms in the geometric progression x³, x⁵, x⁷, ……. (If x ≠ ±1)

Evaluate Σ_k=1¹¹ (2 + 3^k)

The sum of first three terms of a G.P. is  and their product is 1. Find the common ratio and the terms.

How many terms of G.P. 3, 3², 3³ ... are needed to give the sum 120?

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128.

Determine the first term, the common ratio and the sum to n terms of the G.P.

Given a G.P. with a = 729 and 7^th term 64, determine S₇.

Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

If the 4^th, 10^th and 16^th terms of a G.P. are x, y and z, respectively. Prove that x, y, z is in G.P.

Find the sum to n terms of the sequence, 8, 88, 888, 8888...

Find the sum of the products of the corresponding terms of the sequences                                 

2, 4, 8, 16, 32 and 128, 32, 8, 2,  .

Show that the products of the corresponding terms of the sequences form a, ar, ar², …., ar^n-1 and a, AR, AR², …., AR^n-1 a G.P,

and find the common ratio.

Find four numbers forming a geometric progression in which third term is greater than the first term by 9,

and the second term is greater than the 4^th by 18.

If p^th, q^th and r^th terms of a G.P. are a, b and c respectively. Prove that a^{q – r} × b^{r – p} × c^{p – q} = 1.

If the first and the n^th term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P² = (ab)ⁿ

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1) ^th to (2n) ^th term is (1 ÷ rⁿ).

If a, b, c and d are in G.P. show that: (a² + b² + c²) (b² + c² + d²) = (ab + bc – cd)²

Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Find the value of n so that (aⁿ⁺¹ + bⁿ⁺¹) ÷ (aⁿ + bⁿ) may be the geometric mean between a and b.

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3 + 2√2): (3 - 2√2).

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± √ {(A + G) (A - G)}

The number of bacteria in a certain culture doubles every hour.

If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ......n^th term, a_n = n (n +1) 

Find the sum to n terms of the series 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + ...

Find the sum to n terms of the series 3 × 1² + 5 × 2² + 7 × 3² + ...........

Find the sum to n terms of the series {1 ÷ (1 × 2)} + {1 ÷ (2 × 3)} + {1 ÷ (3 × 4)} + …………… 

Find the sum to n terms of the series 5² + 6² + 7² + ... + 20²

Find the sum to n terms of the series 3 × 8 + 6 × 11 + 9 × 14 +........

Find the sum to n terms of the series 1² + (1² + 2²) + (1² + 2² + 3²) + ........

Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4).

Find the sum to n terms of the series whose n^th terms is given by n² + 2n

Find the sum to n terms of the series whose n^th terms is given by (2n – 1)²

Show that the sum of (m + n) ^th and (m – n) ^th terms of an A.P. is equal to twice the m^th term.

If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

Let the sum of n, 2n, 3n terms of an A.P. be S₁, S₂ and S₃, respectively, show that: S₃ =3 (S₂– S₁)

Find the sum of all numbers between 200 and 400 which are divisible by 7.

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

Find the sum of all two-digit numbers which when divided by 4, yields 1 as remainder.

If f is a function satisfying f (x + y) = f(x). f(y) for all x, y belongs to N, such that

f (1) = 3 and i=  = 120, find the value of n.

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively.

Find the last term and the number of terms.

The first term of a G.P. is 1. The sum of the third term and fifth term is 90.

Find the common ratio of G.P.

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order,

we obtain an arithmetic progression. Find the numbers.

A G.P. consists of an even number of terms.

If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112.

If its first term is 11, then find the number of terms.

If (a + bx) ÷ (a - bx) = (b + cx) ÷ (b - cx) = (c + dx) ÷ (c - dx) (x ≠ 0) then show that a, b, c and d are in G.P.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ

The p^th, q^th and r^th terms of an A.P. are a, b, c respectively.

Show that (q - r)a + (r - p)b + (p - q)c = 0

If a (  + ), b (  + ), c (  + ) are in A.P., prove that a, b, c is in A.P.

If a, b, c, d is in G.P, prove that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P.

If a and b are the roots of x² – 3x + p = 0 and c, d are roots of x² – 12x + q = 0,

where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17: 15.

The ratio of the A.M and G.M. of two positive numbers a and b, is m: n.

Show that a: b = [m + √ (m² – n²)]: [m + √ (m² – n²)] 

If a, b, c is in A.P; b, c, d is in G.P and , ,  are in A.P. prove that a, c, e is in G.P.

Find the sum of the following series up to n terms:

(1) 5 + 55 + 555 + …….                       

(2) .6 +.66 +. 666 +………….

Find the 20^th term of the series 2 × 4 + 4 × 6 + 6 × 8 + … + n terms.

Find the sum of the first n terms of the series: 3 + 7 + 13 + 21 + 31 + …

If S₁, S₂, S₃ are the sum of first n natural numbers, their squares and their cubes,

respectively, show that 9S₂² = S₃(1 + 8S₁)

Find the sum of the following series up to n terms: 

(1³÷ 1) + {(1³ + 2³) ÷ (1 + 3)} + {(1³ + 2³ + 3³) ÷ (1 + 3 + 5)} + ………...

Show that {1 × 2² + 2 × 2³ + …...n × (n + 1)²} ÷ {1² × 2 + 2² × 2 + …...n² × (n + 1)} = (3n + 5) ÷ (3n + 1)

A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in

annual installments of Rs 500 plus 12% interest on the unpaid amount.

How much will be the tractor cost him?

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance

in annual installment of Rs 1000 plus 10% interest on the unpaid amount.

How much will the scooter cost him? 

A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four

different persons with instruction that they move the chain similarly.

Assuming that the chain is not broken and that it costs 50 paise to mail one letter.

Find the amount spent on the postage when 8th set of letters is mailed.

A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually.

Find the amount in 15^th year since he deposited the amount and also calculate the total amount after 20 years.

A manufacturer reckons that the value of a machine, which costs him Rs 15625,

will depreciate each year by 20%. Find the estimated value at the end of 5 years.

150 workers were engaged to finish a job in a certain number of days.

4 workers dropped out on second day, 4 more workers dropped out on third day and so on.

It took 8 more days to finish the work. Find the number of days in which the work was completed.