Question

If twice the ${11^{th}}$ term of an A.P is equal to $7$ times of its ${21^{st}}$ term, then its ${25^{th}}$ term is equal to:
A. $24$
B. $120$
C. $0$
D. None of these

Answer 1

The given problem involves the concepts of arithmetic progression. We are given some conditions on the AP according to which we have to find the ${25^{th}}$ term of arithmetic progression. For finding out the nth term of an arithmetic progression, we must know the formula for the general term in an AP: ${a_n} = a + \left( {n - 1} \right)d$.

Complete step by step answer:
So, we have an arithmetic progression in which twice the ${11^{th}}$ term of an A.P is equal to $7$ times of its ${21^{st}}$ term. We know that the difference of any two consecutive terms of an arithmetic progression is constant. So, let us assume the first term of the progression as a and common difference as $d$.Now, we know that the general term of an AP is given by the formula:
${a_n} = a + \left( {n - 1} \right)d$.
So, we calculate the expression for the ${11^{th}}$ term and the ${21^{st}}$ term of the series.
So, ${a_{11}} = a + \left( {11 - 1} \right)d = a + 10d$
Also, ${a_{21}} = a + \left( {21 - 1} \right)d = a + 20d$

Now, we are given the condition that $2\left( {{a_{11}}} \right) = 7\left( {{a_{21}}} \right)$.
So, we get, $2\left( {a + 10d} \right) = 7\left( {a + 20d} \right)$
Opening the brackets, we get,
$\Rightarrow 2a + 20d = 7a + 140d$
Taking all the terms consisting of a to left side of the equation and all terms consisting of d to right side of equation, we get,
$\Rightarrow 2a - 7a = 140d - 20d$
$\Rightarrow - 5a = 120d$
Dividing both sides of equation by $- 5$, we get,
$\Rightarrow a = \left( {\dfrac{{120}}{{ - 5}}} \right)d$

Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow a = - 24d$
Now, taking all terms to left side of equation, we get,
$\Rightarrow a + 24d = 0 - - - - \left( 1 \right)$
Now, we have to find the ${25^{th}}$ term of the arithmetic progression.
We know the formula for the general term of AP as: ${a_n} = a + \left( {n - 1} \right)d$.
So, ${a_{25}} = a + \left( {25 - 1} \right)d$
$\therefore {a_{25}} = a + 24d$
Now, from equation $\left( 1 \right)$, we know the value of the expression $a + 24d$. So, we get,
$\therefore {a_{25}} = 0$

Hence, the option C is the correct answer.

Note: Arithmetic progression is a series where any two consecutive terms have the same difference between them. The common difference of an arithmetic series can be calculated by subtraction of any two consecutive terms of the series. Any term of an arithmetic progression can be calculated if we know the first term and the common difference of the arithmetic series as: ${a_n} = a + \left( {n - 1} \right)d$. Take care while handling the steps involving tedious calculations.