Question

What is $x$ if the sequence $1,5,2x+3...$ is an arithmetic sequence?

Answer 1

Find the value of $x$ from the sequence of $1,5,2x+3...$

Given, that the arithmetic sequence is $1,5,2x+3...$

If the three terms $T_1,T_2,\mathrm{and}{T}_3$ are in an arithmetic sequence, then the common difference between the consecutive terms is equal.

So,

$$\begin{gathered} T_3-T_2=T_2-T_1 \\ \Rightarrow (2x+3)-5=5-1\left\{\mathrm{Let}{T}_1=1,T_2=5,T_3=2x+3\right\} \\ \Rightarrow 2x=4-3+5 \\ \Rightarrow 2x=6 \\ \Rightarrow x=3 \end{gathered}$$

The obtained sequence for $1,5,2x+3...$is $1,5,9$ and the common difference is $4$.

Hence, the value of $x$ for the sequence $1,5,2x+3...$ is $3$.