Question

find T100 and S50 if the sum of 4 terms of an A.P is 16 that of 20 terms is 400​

Answer 1

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$\large\bold\red{Given :- }$

Sum of 4 terms is 16

Sum of 20 terms is 400

$\large\bold\red{To find :-}$

■50th term of an AP

■Sum of 100 terms

$\large\bold\red{Formula \: used :-}$

$\small\bold\red{Sum \: of \: n \: terms \: of \: ap \: = \frac{n}{2}(2a + (n - 1) \times d) } \\ \small\bold\red{nth \: term \: of \: an \: ap \: = a + (n - 1) \times d}$

$\large\bold\blue{Solution:- }$

$Sum \: of \: first \: 4 \: terms \: = 16 \\ \frac{4}{2} (2a + (4 - 1) \times d) = 16 \\ 2a + 3d = 8......(1) \\ \\ Sum \: of \: 20 \: terms \: = 400 \\ \frac{20}{2} (2a + (20 - 1) \times d) = 400 \\ 2a + 19d = 40......(2) \\ \\ \large\bold\red{(subtracting \: (1) \: from \: (2))} \\ 2a + 19d - 2a - 3d = 40 - 8 \\ 16d \: = 32 \\ = > \: d = 2 \\ \large\bold\red{put \: d = 2 \: in \: equation \: (1)} \\ 2a + 3 \times 2= 8 \\ 2a = 8 - 6 \\ 2a = 2 \\ = > \: a \: = 1 \\ so \: 100th \: term \: = a + (100 - 1) \times d \\ = 1 + 99 \times 2 \\ = 1 + 198 \\ = 199 \\ Sum \: of \: 50 \: terms \: = \frac{50}{2} (2 \times 1 \: + (50 - 1) \times 2) \\ = 25 \times (2 + 98) \\ = 25 \times 100 \\ = 2500$