Question

If the first two terms of a H.P. are $2/5$ and $12/13$ respectively. Then find the largest term ?

Answer 1

Given the first two terms of H.P is $\frac{2}{5}\text{and}\frac{12}{13}$

which means

$\frac{1}{a}=\frac{2}{5}\Rightarrow a=\frac{5}{2}and,\frac{1}{a+d}=\frac{12}{13}\Rightarrow d=\frac{-17}{12}\therefore a_n=a+(n-1)d=\frac{5}{2}+(n-1)\frac{-17}{12}=\frac{47-17n}{12}\Rightarrow \frac{1}{a_n}=\frac{12}{47-17n}\text{for}\frac{1}{a_n}\text{(to be largest term},(47-17n)\text{should be smallest positive interger}\text{for}n=2,(47-17n)=13\text{for}n=1,(47-17n)=30,\text{term can't be taken, since in question it is already be seen that}\frac{2}{5}<\frac{12}{13}\text{for}n=3,4,....,\text{the term}(47-17n)\text{will be negative}\therefore \text{largest term in the given}H.P\text{is}\frac{12}{13}$