Question

In an arithmetic progression of 50 terms , the sum of first ten terms is 210 and the sum of last fifteen terms is 2565 . Find the arithmetic progression .

Answer 1

Let a be the first term and d be the common difference of the given AP.
Sum of the first n terms is given by
Sn = n/2 {2a + (n - 1)d}
Putting n = 10, we get
S₁₀ = 10/2 {2a + (10 - 1)d}
210 = 5 (2a + 9d)
2a + 9d = 210/5
2a + 9d = 42 ...............(1)
Sum of the last 15 terms is 2565
⇒ Sum of the first 50 terms - sum of the first 35 terms = 2565
S₅₀ - S₃₅ = 2565
⇒ 50/2 {2a + (50 - 1)d} - 35/2 {2a + (35 - 1)d} = 2565
25 (2a + 49d) - 35/2 (2a + 34d) = 2565
⇒ 5 (2a + 49d) - 7/2 (2a + 34d) = 513
⇒ 10a + 245d - 7a + 119d = 513
⇒ 3a + 126d = 513
⇒ a + 42d = 171 ........(2)
Multiply the equation (2) with 2, we get
2a + 84d = 342 .........(3)
Subtracting (1) from (3)
2a + 84d = 342
2a + 9d = 42
- - -
_______________
75d = 300
_______________
d= 4
Now, substituting the value of d in equation (1)
2a + 9d = 42
2a + 9*4 = 42
2a = 42 - 36
2a = 6
a = 3
So, the required AP is 3, 7, 11, 15, 19, 23, 27, 31, 35, 39 .....