(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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Class 11

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Sequences and Series

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Arithmetic progression

If $\log 2$,$\log (2^x - 1)$,and $\log (2^x + 3)$ are in arithmetic progression,then the value of $x$ is:

Medium

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The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$,find the number of the sides of the polygon.

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Given that $n$ $A$.$M$.'s are inserted between two sets of numbers $a, 2b$ and $2a, b$,where $a, b \in R$. Suppose further that the $m^{th}$ mean between these sets of numbers is the same,then the ratio $a:b$ equals

Medium

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If the first,second and last terms of an $A.P.$ are $a, b$ and $2a$ respectively,then its sum is:

Easy

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Four numbers are in an arithmetic progression. The sum of the first and last terms is $8$ and the product of the two middle terms is $15$. What is the smallest number in the sequence?

Medium

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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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If $\log 2$,$\log (2^x - 1)$,and $\log (2^x + 3)$ are in arithmetic progression,then the value of $x$ is:

The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$,find the number of the sides of the polygon.

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