The numbers 1,5 and 25 can be three terms (not necessarily...

Given, numbers $1,5,25$ Let $\mathrm{p}^{\text{th }}$ term $=1 \Rightarrow \mathrm{a}+(\mathrm{p}-1) \mathrm{d}=1$...(1) Let $\mathrm{q}^{\text{th }}$ term $=5 \Rightarrow \mathrm{a}+(\mathrm{q}-1) \mathrm{d}=5$...(2) Let $r^{\text{th }}$ term $=25 \Rightarrow a+(r-1) d=25$...(3) Applying (3) - (2) $\Rightarrow \mathrm{r}-\mathrm{q}=25-5=20$ Applying (2) $-(1) \Rightarrow \mathrm{q}-\mathrm{p}=5-1=4\frac{r-q}{q-p}=\frac{20}{4}=5$ which is an integer. So, the given series forms an AP. Infinite AP's are possible.