(A) Let the first term be $a$ and the common difference be $d$. The $n^{th}$ term is given by $a_n = a + (n-1)d$.According to the problem,$100 \times

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(A) Let the first term be $a$ and the common difference be $d$. The $n^{th}$ term is given by $a_n = a + (n-1)d$.According to the problem,$100 \times a_{100} = 50 \times a_{50}$.$100(a + 99d) = 50(a + 49d)$.Dividing by $50$,we get $2(a + 99d) = a + 49d$.$2a + 198d = a + 49d$.$a = 49d - 198d = -149d$.The $150^{th}$ term is $a_{150} = a + 149d$.Substituting $a = -149d$,we get $a_{150} = -149d + 149d = 0$.

Class 11 Mathematics Sequences and Series Arithmetic progression

Medium

If $100$ times the $100^{th}$ term of an Arithmetic Progression with a non-zero common difference is equal to $50$ times its $50^{th}$ term,then what is its $150^{th}$ term?

A
$0$
B
$-150$
C
$150$ times its $50^{th}$ term
D
$150$

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Sequences and Series

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Arithmetic progression

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DifficultJEE MAIN 2025

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If the sum of the first $10$ terms of an arithmetic progression is $4$ times the sum of its first $5$ terms,then the ratio of its first term to the common difference is......

Medium

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Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15:7$,then $S_{15} - S_5$ is equal to:

DifficultJEE MAIN 2024

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If $100$ times the $100^{th}$ term of an Arithmetic Progression with a non-zero common difference is equal to $50$ times its $50^{th}$ term,then what is its $150^{th}$ term?

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