Question

Find the number of terms of the AP $54,51,48,......$ so that their sum is $513$.

Answer 1

Step 1. Find the common difference

Arithmetic progression or sequence is a mathematical sequence in which the difference between two consecutive terms is always a constant. It is represented as A. P.

The difference between any two consecutive term is called as a common difference and it is denoted by $\mathrm{d}$.

So the common difference is

$$\begin{gathered} \mathrm{d}=51-54 \\ =-3 \end{gathered}$$

So the common difference is $\mathrm{d}=-3$

Step 2. Find the number of terms

Formula to find the sum of first ${\mathrm{n}}^{\mathrm{th}}$ terms of an AP is,

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}]$

Where $\mathrm{a}$ is the first term, $\mathrm{n}$ is the number of terms in the sequence and $\mathrm{d}$ is common difference.

Given ${\mathrm{S}}_{\mathrm{n}}=513$

$$\begin{gathered} \therefore 513=\frac{n}{2}\left[2a+d\left(n-1\right)\right] \\ \Rightarrow 513=\frac{n}{2}\left[2·54-3\left(n-1\right)\right] \\ \Rightarrow 1026=n\left[108-3n+3\right] \\ \Rightarrow 3n^2-111n+1026=0 \\ \Rightarrow n^2-37n+342=0 \\ \Rightarrow n^2-18n-19n-342=0 \\ \Rightarrow n\left(n-18\right)-19\left(n-18\right)=0 \\ \Rightarrow \left(n-18\right)\left(n-19\right)=0 \end{gathered}$$

$\therefore n=18$ or $n=19$

Hence, the number of terms is either $18$ or $19$, so that their sum is $513$.