Question

How many terms of the AP: 9, 17, 25,….must be taken to give a sum of 636?

Answer 1

Hint: Use the formula of sum of n terms of AP, which is as follows: \[\dfrac{n}{2}\left[ 2a+(n-1)d \right]\]. Here n is the number of terms of the AP, a is the first term of the AP and d is the common difference of the AP.

Complete step-by-step solution -
In the question, we have to find the number of terms of the AP: 9, 17, 25,….that must be taken to give a sum of 636.
Now, according to the formula of the sum of the n terms of AP, we have: \[\dfrac{n}{2}\left[ 2a+(n-1)d \right]\].
The n is the number of terms, a is the first term and d is the common difference of the AP.
Now the first series of the AP is 9, so we have \[a=9\]. Common difference is the difference in the second and the first terms and that is \[d=17-9=8\]. Finally, n is the required number of terms of the AP.
So we have the sum given as 636, so according to the sum formulas, we have:
\[\begin{align}
  & \Rightarrow \dfrac{n}{2}\left[ 2a+(n-1)d \right]=636 \\
 & \Rightarrow \dfrac{n}{2}\left[ 2\times 9+(n-1)8 \right]=636 \\
 & \Rightarrow n\left( 18+8\left( n-1 \right) \right)=1272 \\
 & \Rightarrow 8{{n}^{2}}+10n-1272=0 \\
\end{align}\]
So, here we get a quadratic equation, of the form \[a{{x}^{2}}+bx+c=0\], which is solved by using the formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Now, we have \[a=8,b=10,c=-1272\]
So the value of n is as follows:
\[\begin{align}
  & \Rightarrow 8{{n}^{2}}+10n-1272=0 \\
 & \Rightarrow n=\dfrac{-10\pm \sqrt{{{10}^{2}}-4\cdot \;8\left( -1272 \right)}}{2\cdot \;8} \\
 & \Rightarrow n=\dfrac{-10\pm \sqrt{40804}}{16} \\
  & \Rightarrow \dfrac{-10\pm 202}{16} \\
 & \Rightarrow n=-\dfrac{212}{16},\,n=\dfrac{192}{16} \\
 & \Rightarrow n=-\dfrac{53}{4},\,\,n=12 \\
\end{align}\]
Now, the number of terms of the AP can’t be negative so we will just take the positive value of n.
Finally, the required number of terms of the AP is \[n=\text{12}\].

Note: When finding the common difference of the Arithmetic sequence, we need to be careful that it is the difference in the two consecutive terms. For example, the common difference is always the second term minus the first terms and not the first term minus the second term. If the common difference is negative, then the AP is decreasing else it is increasing.