Question:
Question 3: Let Sn denote the sum of n terms of an A.P. whose first term is a . If the common difference d is given by d = Sn - kSn-1 Sn-2, then k =
Answer:
Let a is the first term, n is the number of terms and d is the common difference of AP.
Given d = Sn - kSn-1 + Sn-2 and we have to find the value of k.
Now let n = 3
So AP is : a , a + d, a + 2d
and d = S3 - kS3-1 + S3-2
=> d = S3 - kS2 + S1 ..............1
Sum of n terms of an AP is given as:
Sn = (n/2)*{2a + (n-1)d}
Now S1 = a
S2 = (2/2)*(2a + (2-1)d) (n =2)
=> S2 = (2a + d)
S3 = (3/2)*(2a + (3-1)d) (n =3)
=> S3 = (3/2)*(2a + 2d)
=> S3 = 3(a + d)
=> S3 = 3a + 3d
Put value of S1 , S2 and S3 in equation 1, we get
d = 3a + 3d - k(2a + d) + a
=> d = 4a + 3d - k(2a + d)
=> k(2a + d) = 4a + 3d - d
=> k(2a + d) = 4a + 2d
=> k(2a + d) = 2(2a + d)
=> k = 2
So value of K is 2
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