Question

Find $\mathrm{a},\mathrm{b}$ and $\mathrm{c}$such that the following numbers are in AP: $\mathrm{a},7,\mathrm{b},23,\mathrm{c}.$

Answer 1

Step 1: Find common difference of AP

AP = $\mathrm{a},7,\mathrm{b},23,\mathrm{c}$

Here, common difference is

Common difference ($\mathrm{d}$) = $7-\mathrm{a}=\mathrm{b}-7=23-\mathrm{b}=\mathrm{c}-23$

Now, simplification of all equations.

$$\begin{gathered} \Rightarrow \mathrm{b}-7=23-\mathrm{b} \\ \Rightarrow 2\mathrm{b}=23+7=30 \\ \Rightarrow \mathrm{b}=15 \end{gathered}$$

Step 2: Find $\mathbf{a}\mathbf{}$and $\mathbf{c}$

$7-\mathrm{a}=\mathrm{b}-7$

$$\begin{gathered} \Rightarrow 7-a=15-7 \\ \Rightarrow a=-1 \end{gathered}$$

and,

$$\begin{gathered} \Rightarrow c-23=23-b=23-15 \\ \Rightarrow c=46-15=31 \end{gathered}$$

Hence, the value of $\mathit{a}\mathbf{,}\mathbf{}\mathit{b}\mathbf{}\mathbf{\text{and}}\mathbf{}\mathit{c}$ are $\mathbf{–}\mathbf{}\mathbf{1}\mathbf{,}\mathbf{15}\mathbf{}\mathbf{\text{and}}\mathbf{}\mathbf{31}$respectively.