Question

Sum of the first 20 terms of an A.P is $-240$ and its first term is given by 7. Find the ${{24}^{th}}$ term of the A.P.
(a) $-24$
(b) $-39$
(c) 1
(d) 0

Answer 1

In this question, We are given that the sum of 20 terms of an A.P is $-240$. Now using the formula ${{S}_{n}}=\dfrac{n}{2}\,\left[ 2a+\left( n-1 \right)d \right]$ for calculating the sum of first $n$ terms of an A.P we will determine the value of the common difference $d$. Then we will substitute the value of $d$ in the formulate to calculate the ${{n}^{th}}$ terms of an A.P denotes by ${{a}_{n}}$ which is given by
${{a}_{n}}=a+\left( n-1 \right)d$ with $n=24$ to get the ${{24}^{th}}$ term of the A.P.

Complete step-by-step answer:
We are given that the sum of 20 terms of an A.P is $-240$.
Also the first term of the arithmetic progression is given as 7.
Now we know that the sum of first $n$ terms of an A.P is given by
${{S}_{n}}=\dfrac{n}{2}\,\left[ 2a+\left( n-1 \right)d \right].....(1)$
Where
${{S}_{n}}$ denotes the sum of first $n$ terms.
$a$ denotes the first term of the A.P.
$d$ denotes the common difference.
Now on comparing the variable with out given information, we get
$n=20$, $a=7$ and ${{S}_{n}}=-240$
We will now calculate the common difference by substituting the value $n=20$, $a=7$ and ${{S}_{n}}=-240$ in (1),

& -240=\dfrac{20}{2}\,\left[ 2\left( 7 \right)+\left( 20-1 \right)d \right] \\\ & \Rightarrow -240=10\,\left[ 14+\left( 19 \right)d \right] \\\ \end{aligned}$$ On dividing the above equation by 10, we get $$-24=\,14+\left( 19 \right)d$$ Now on simplify the above equation to find the value of $$d$$, we have $$\begin{aligned} & -24-14=\,\left( 19 \right)d \\\ & \Rightarrow 19d=-38 \\\ & \Rightarrow d=\dfrac{-38}{19} \\\ & \Rightarrow d=-2 \\\ \end{aligned}$$ Thus we get that the common difference $$d$$ of the given A.P is $$-2$$. Now we know that the $${{n}^{th}}$$ terms of an A.P denotes by $${{a}_{n}}$$ is given by $${{a}_{n}}=a+\left( n-1 \right)d...........(2)$$ Thus, in order to find the $${{24}^{th}}$$ term of the A.P, we will now substitute the values $$n=24$$, $$a=7$$ and $$d=-2$$ in (2). Then we get, $$\begin{aligned} & {{a}_{24}}=7+\left( 24-1 \right)\left( -2 \right) \\\ & =7+23\left( -2 \right) \\\ & =7-46 \\\ & =-39 \end{aligned}$$ Hence we get that the $${{24}^{th}}$$ term of the A.P is $$-39$$. **So, the correct answer is “Option b”.** **Note:** In this problem, we have to carefully use the formulas for the sum of first $$n$$ terms of an A.P and to find the $${{n}^{th}}$$ terms of an A.P in proper order. We can also find an equation for the $${{n}^{th}}$$ terms of the A.P in terms of common difference and then substituting the value to get the desired value.