Question

The $26{\text{th}}$, $11{\text{th}}$ and the last term of an A.P. are $0$, $3$ and $\dfrac{{ - 1}}{5}$ respectively. Find the common difference and number of terms.

Answer 1

In this question, we will use the $n{\text{th}}$ term formula to find the first term and common difference of A.P. then substitute the obtained values in the equation formed for the last term to find the numbers of terms.

Formula Used: We know that the formula for $n{\text{th}}$ term formula of an A.P. series is ${a_n} = a + \left( {n - 1} \right)d$. Here, ${a_n}$ is the $n{\text{th}}$ term, $a$ is the first term, $d$ is the common difference and $n$ is the number of terms.

Complete step-by-step answer:
We know that $26{\text{th}}$ term of the given A.P. is $0$. Substitute the values in the $n{\text{th}}$ term formula for the $26{\text{th}}$ term.
$\Rightarrow$ $0 = a + \left( {26 - 1} \right)d$
$\Rightarrow$ $0 = a + 25d\;\;...\left( 1 \right)$
We know that $11{\text{th}}$ term of the given A.P. is $3$. Substitute the values in the $n{\text{th}}$ term formula for the $11{\text{th}}$ term.
$\Rightarrow$ $3 = a + \left( {11 - 1} \right)d$
$\Rightarrow$ $3 = a + 10d\;\;...\left( 2 \right)$
Subtract equation $\left( 2 \right)$ from equation $\left( 1 \right)$ and solve.
$\Rightarrow$ $- 3 = a + 25d - a - 10d$
$\Rightarrow$ $- 3 = 15d$
$\Rightarrow$ $\dfrac{{ - 3}}{{15}} = d$
$\Rightarrow$ $\dfrac{{ - 1}}{5} = d$
Now, substitute the common difference in equation $\left( 1 \right)$.
$\Rightarrow$ $0 = a + 25\left( {\dfrac{{ - 1}}{5}} \right)$
$\Rightarrow$ $0 = a - 5$
$\Rightarrow$ $a = 5$
We know that the last term of A.P. is $\dfrac{{ - 1}}{5}$. Substitute all the values in $n{\text{th}}$ term formula of the A.P. series and calculate the number of terms.
$\dfrac{{ - 1}}{5} = 5 + \left( {n - 1} \right)\left( {\dfrac{{ - 1}}{5}} \right)$
$\Rightarrow - 1 = 25 + \left( {n - 1} \right)\left( { - 1} \right)$
$\Rightarrow - 1 = 25 - n + 1$
$\Rightarrow n = 27$
Therefore, the common difference is $\dfrac{{ - 1}}{5}$ and the number of terms of A.P. is $27$.

Note: In this question, we need to find the values of first term and common difference. Put these values in the equation obtained by the $n{\text{th}}$ term formula of A.P for the last term of A.P. to find the number of terms.