Let a 1 , a 2 , a 3 , … be an arithmetic progression with a 1 = 7 and common difference 8 . Let T 1 , T 2 , T 3 , … be such that T 1 = 3 and T n + 1 - T n = a n fo n ≥ 1 . Then, which of the following is/are TRUE?
C. T 30 = 3454
Given, a n = 7 + n - 1 8 and T 1 = 3 Also T n + 1 = T n + a n Now T n = T n - 1 + a n - 1 Putting n = 1 we get, T 2 = T 1 + a 1 Now putting n = 2 we get, T 3 = T 2 + a 2 ⇒ T 3 = T 1 + a 1 + a 2 And so on we get, ⇒   T n + 1 = T 1 + a 1 + a 2 + … + a n ⇒   T n + 1 = T 1 + n 2 2 7 + n - 1 8 ⇒   T n + 1 = T 1 + n 4 n + 3           ⋯ 1 Now for n = 19 we get, T 20 = 3 + 19 79 = 1504 And for n = 29 we get, T 30 = 3 + 29 119 = 3454 option C is correct Now finding sum we get, ∑ k = 1 20 T k = 3 + ∑ k = 2 20 T k = 3 + ∑ k = 1 19 3 + 4 n 2 + 3 n = 3 + 3 19 + 3 19 20 2 + 4 19 20 39 6 = 3 + 10507 = 10510 option B is correct And Similarly ∑ k = 1 30 T k = 3 + ∑ k = 1 29 4 n 2 + 3 n + 3 = 35615
Related: Mathematics — Sequences and Series · All PYQ Banks