Question

How many terms of the AP $-6,\,\,\dfrac{-11}{2},\,-5..........$ are needed to give the sum of -25?

Answer 1

Hint: Here, we will use the formula for the sum of n terms of an AP. We will take the number of required terms as a variable, say n and then we will calculate the value of n by using the formula.

Complete step-by-step answer:
We know that a sequence in which the difference between the two consecutive terms is always constant is an AP. This constant is termed as the common difference of the AP.
If the first term of an AP be ‘a’ and its common difference be ‘d’, then the sum of n terms of the AP is given by the formula:
${{S}_{n}}=\dfrac{n}{2}\left\{ 2a+\left( n-1 \right)d \right\}...........\left( 1 \right)$
Since, the AP given in this problem is:
 $-6,\,\,\dfrac{-11}{2},\,-5..........$

Let us take the required number of terms of this AP to give a sum of -25 be n.
Common difference of this AP is:
$\begin{align}
  & d=\dfrac{-11}{2}-\left( -6 \right) \\
 & \,\,\,\,=\dfrac{-11+12}{2} \\
 & \,\,\,\,=\dfrac{1}{2} \\
\end{align}$
Also, the first term of this AP is -6.

Putting the values of the first term and the common difference of this AP in equation (1), we get:
$\begin{align}
  & {{S}_{n}}=\dfrac{n}{2}\left\{ 2\times \left( -6 \right)+\left( n-1 \right)\dfrac{1}{2} \right\} \\
 & \,\,\,\,\,\,=\dfrac{n}{2}\left\{ -12+\dfrac{\left( n-1 \right)}{2} \right\} \\
\end{align}$

Since, it is given that the sum is = -25. So, on substituting this value in above equation, we get:
$\begin{align}
  & -25=\dfrac{n}{2}\left( \dfrac{-24+n-1}{2} \right) \\
 & -25=\dfrac{n\left( n-25 \right)}{4} \\
 & -25\times 4={{n}^{2}}-25n \\
 & {{n}^{2}}-25n+100=0 \\
\end{align}$

Here, we get a quadratic equation in n. So, we will have 2 values of n on solving this quadratic equation.
On solving this quadratic equation using factorization method, we get:
$\begin{align}
  & {{n}^{2}}-5n-20n+100=0 \\
 & n\left( n-5 \right)-20\left( n-5 \right)=0 \\
 & \left( n-5 \right)\left( n-20 \right)=0 \\
\end{align}$
So, from here we get two values of n as:
n = 5 or n = 20
Hence, the number of required terms of this given AP to give a sum of -25 is either 5 or 20.

Note: It should be noted that since we have obtained two values of n, it means that the sum of the terms from the 6^th to 12^th place is zero. Students should use the formula for the sum of n terms of an AP carefully to avoid mistakes.