Question

If 100 times the ${100^{th}}$term of an A.P with non-zero common difference equal the 50 times its ${50^{th}}$term, then the ${150^{th}}$ term this A.P is
$\left( a \right)$ - 150
$\left( b \right)$ 150 times its ${50^{th}}$ term
$\left( c \right)$ 150
$\left( d \right)$ 0

Answer 1

In this particular question use the concept that the ${n^{th}}$ term of an A.P is given as ${a_n} = a + \left( {n - 1} \right)d$, where ${a_n}$ is the ${n^{th}}$ term, a is the first term and d is the common difference, so use these concepts to reach the solution of the question.

Complete step-by-step answer :
Given data:
100 times the ${100^{th}}$term of an A.P with non-zero common difference equal the 50 times its ${50^{th}}$term.
Now as we know that the ${n^{th}}$ term of an A.P is given as ${a_n} = a + \left( {n - 1} \right)d$, where ${a_n}$ is the ${n^{th}}$ term, a is the first term and d is the common difference.
So the ${100^{th}}$ term of an A.P is
$\Rightarrow {a_{100}} = a + \left( {100 - 1} \right)d = a + 99d$
And the ${50^{th}}$ term of an A.P is
$\Rightarrow {a_{50}} = a + \left( {50 - 1} \right)d = a + 49d$
Now according to the question 100 times the ${100^{th}}$term of an A.P is equal the 50 times its ${50^{th}}$term.
$\Rightarrow 100\left( {{a_{100}}} \right) = 50\left( {{a_{50}}} \right)$
Now substitute the values we have,
$\Rightarrow 100\left( {a + 99d} \right) = 50\left( {a + 49d} \right)$
Now simplify it we have,
$\Rightarrow \dfrac{{100}}{{50}}\left( {a + 99d} \right) = \left( {a + 49d} \right)$
$\Rightarrow 2\left( {a + 99d} \right) = \left( {a + 49d} \right)$
$\Rightarrow 2a + 198d = a + 49d$
$\Rightarrow a + 149d = 0$................ (1)
Now we have to find out the ${150^{th}}$ term this A.P.
So, the ${150^{th}}$ term of an A.P is
$\Rightarrow {a_{150}} = a + \left( {150 - 1} \right)d = a + 149d$
Now from equation (1) we have,
$\Rightarrow {a_{150}} = a + 149d = 0$
So, the ${150^{th}}$ term of this A.P is zero.
So this is the required answer.
Hence option (d) is the correct answer.

Note : Whenever we face such types of questions the key concept we have to remember is that always recall formula of ${n^{th}}$ term of an A.P which is stated above, then using this formula find out the ${100^{th}}{\text{ and }}{50^{th}}$ term as above, then according to question equate them as above and find the condition we will get the required answer.