Question

If the first, second and last terms of an Arithmetic series are$a,b$ and $c$ respectively, then the number of terms is

Answer 1

Step 1: Solve for the common difference of the given series

Given that,

first term $=a$

second term$=b$

common difference $d=$ second term $-$ first term $=$$b-a$

Step 2: Solve for the number of terms

Given that Last term $l=c$ and say, $n$is the number of terms in the series

$\begin{array}{crcl}\Rightarrow & a+\left(n-1\right)d& =& c\\ \Rightarrow & \left(n-1\right)\left(b-a\right)& =& c-a\\ \Rightarrow & n\left(b-a\right)& =& \left(c-a\right)+\left(b-a\right)\\ \Rightarrow & n& =& \frac{b+c-2a}{\left(b-a\right)}\end{array}$

Hence, number of terms is $\frac{b+c-2a}{b-a}$