Question:
The ratio of the 5th and 3rd terms of an AP is 2:5. find the ratio of the 15th and 7th terms.
Answer:
Let a is the first term and d is the common difference of the AP
Now, given T5 : T3 = 2 : 5
=> T5 / T3 = 2 / 5
=> {a + (5 - 1)d}/{a + (3 - 1)d} = 2/5
=> {a + 4d}/{a + 2d} = 2/5
=> 5{a + 4d} = 2{a + 2d}
=> 5a + 20d = 2a + 4d
=> 5a - 2a = 4d - 20d
=> 3a = -16d
=> a = -16d/3
Now, T15 / T7 = {a + (15 - 1)d}/{a + (7- 1)d}
=> T15 / T7 = {a + 14d}/{a + 6d}
=> T15 / T7 = {-16d/3 + 14d}/{-16d/3 + 6d}
=> T15 / T7 = {(-16d/2 + 42d)/3}/{(-16d + 18d)/3}
=> T15 / T7 = {26d/3}/{2d/3}
=> T15 / T7 = 26d/2d
=> T15 / T7 = 13/1
=> T15 : T7 = 13 : 1
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