Math, asked by laraibtulip, 3 months ago

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

Answers

Answered by mathdude500
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

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