Question

The solution of the equation $(\mathrm{x}+1)+(\mathrm{x}+4)+(\mathrm{x}+7)+....(\mathrm{x}+28)=155$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A

$1$

The explanation for the correct answer.

Step 1: Find the common difference.

Given: $(\mathrm{x}+1)+(\mathrm{x}+4)+(\mathrm{x}+7)+....(\mathrm{x}+28)=155$

Given equation is in Arithmetic Progression with the initial term as $(x+1)$

Common difference

$$\begin{gathered} \mathrm{d}={\mathrm{a}}_2-{\mathrm{a}}_1 \\ \mathrm{d}=(\mathrm{x}+4)-(\mathrm{x}+1)=3 \end{gathered}$$

Step 2: Find the number of elements in AP.

Let the number of terms in AP are $\mathrm{n}$.

The nth term of AP is given by ${\mathrm{a}}_{\mathrm{n}}=\mathrm{a}+(\mathrm{n}-1)\times \mathrm{d}$

$$\begin{gathered} \Rightarrow \mathrm{x}+28=\mathrm{x}+1+(\mathrm{n}-1)\times 3 \\ \Rightarrow \mathrm{x}+28=\mathrm{x}+1+3\mathrm{n}-3 \\ \Rightarrow 28=3\mathrm{n}-2 \\ \Rightarrow 28+2=3\mathrm{n} \\ \Rightarrow 30=3\mathrm{n} \\ \Rightarrow 10=\mathrm{n} \end{gathered}$$

Step 3: Find the value of $x$

$$\begin{gathered} {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}(\mathrm{first}\mathrm{term}+\mathrm{last}\mathrm{term}) \\ \Rightarrow 155=\frac{10}{2}(\mathrm{x}+1+\mathrm{x}+28) \\ \Rightarrow 155=5\left(2\mathrm{x}+29\right) \\ \Rightarrow \frac{155}{5}=2\mathrm{x}+29 \\ \Rightarrow 31=2\mathrm{x}+29 \\ \Rightarrow 31-29=2\mathrm{x} \\ \Rightarrow 2=2\mathrm{x} \\ \Rightarrow 1=\mathrm{x} \end{gathered}$$

Hence option (a) is the correct answer.