Find the value of $m$ given that the sum of the first $m$ terms is equal to the sum of the first $(m+1)$ terms.

Question

The first term of an arithmetic progression is $100$ and the common difference is $-5$. The answer should be $20$, but how? Please explain the solution.

Answer 1

Solve using the formula for the sum of an arithmetic progression

$$S_m:=\frac m2\left(200-5(m-1)\right)=\frac{m+1}2\left(200-5m\right)=S_{m+1}\iff$$

$$205m-5m^2=195m-5m^2+200\iff10m=200$$

Answer 2

We are told that $s_{m+1}=s_m$, hence $a_{m+1}=0$. But we also have $a_{m+1}=100-m\cdot5\,$. The conclusion is that $m=20$.