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

Hint: 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),
\[\begin{align}
  & -240=\dfrac{20}{2}\,\left[ 2\left( 7 \right)+\left( 20-1 \right)d \right] \\
 & \Rightarrow -240=10\,\left[ 14+\left( 19 \right)d \right] \\
\end{align}\]
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
\[\begin{align}
  & -24-14=\,\left( 19 \right)d \\
 & \Rightarrow 19d=-38 \\
 & \Rightarrow d=\dfrac{-38}{19} \\
 & \Rightarrow d=-2 \\
\end{align}\]
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,
\[\begin{align}
  & {{a}_{24}}=7+\left( 24-1 \right)\left( -2 \right) \\
 & =7+23\left( -2 \right) \\
 & =7-46 \\
 & =-39
\end{align}\]
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.