Question

If $1+4+7+10.....+x=287$, find the value of $x$.

• A. $40$
• B. $39$
• C. $41$
• D. $42$

Answer 1

The correct option is A

$40$

Explanation for the correct option:

Step $1:$ Find the common difference.

Since, the given series is an AP with the first term $a$ as $1$.

Common difference $\left(d\right)$$=4-1$

Therefore, $d=3$.

Step $2:$ Calculate the total number of terms in the given AP.

Apply the formula of the sum of the first $n$ terms of AP.

$$\begin{gathered} 287=\frac{n}{2}\left\{2\times 1+\left(n-1\right)3\right\} \\ \Rightarrow 287=\frac{n}{2}\left\{2+3n-3\right\} \\ \Rightarrow 287=\frac{n}{2}\left\{3n-1\right\} \\ \Rightarrow 574=3n^2-n \\ \Rightarrow 3n^2-n-574=0 \end{gathered}$$

Use the quadratic formula to find the value of $n$.

$$\begin{gathered} n=\frac{1\pm \sqrt{1+4\times 3\times 574}}{2\times 3} \\ \Rightarrow n=\frac{1\pm \sqrt{1+6888}}{6} \\ \Rightarrow n=\frac{1\pm \sqrt{6889}}{6} \\ \Rightarrow n=\frac{1\pm 83}{6} \\ \Rightarrow n=\left\{\frac{84}{6},\frac{-82}{6}\right\} \end{gathered}$$

Since $n$ is the number of terms. So, $n$ cannot be negative.

Therefore, $n=14$

Step $3:$ Calculate the required value.

Since the total number of terms in the given AP is $14$. So, $x$ is the $14th$ term of the AP.

Compute the value of $x$.

$$\begin{gathered} x=1+\left(14-1\right)3 \\ \Rightarrow x=1+13\times 3 \\ \Rightarrow x=1+39 \end{gathered}$$

Therefore, the value of $x$ is $40$.

Hence, option $A$ is the correct answer.