Question

Prove that no matter what the real number $a$ and $b$ are, the sequence with the $nth$ term $(a+nb)$ is always an $AP$. Also find the sum of first $20$ terms.

Answer 1

Let $c=a+b$

$\therefore a+nb=c-b+nb$

$=c+(n-1)b$

Which is the general term for $n^{th}$ term of AP

Sum of first $20$ terms$=\frac{20}{2}[2c+(20-1\left)b\right]$

$=\frac{20}{2}\left[2\right(a+b)+(20-1\left)b\right]$

$=10[2a+2b+20b-b]$

$=10[2a+20b+b]$

$=10[2a+21b]$