Question

If `a_1,a_2,a_3, ,a_n` are an A.P. of non-zero terms, prove that `1/(a_1a_2)+1/(a_2a_3)++1/(a_(n-1)a_n)=`

Answer 1

LHS
`1/(a_1a_2)+1/(a_2a_3)+...+1/(a_(n-11)a_n)`
`1/(a_1(a+1+d))+1/(a_2(a_2+d))+...+1/(a_(n-1)(a_(n-1))+d)`
`1/d(1/a_1-1/(a_1+d))+1/d(1/(a_2+d))`
`1/d(1/a_1-1/a_2+1/a_2-1/a_3+...+1/(a_(n-1))-1/a_n)`
`1/d(1/a_1-1/a_n)`
`1/d((a_n-a_1)/(a_1a_n))`
`1/d*((n-1)d)/(a_1a_n)`
`(n-1)/(a_1a_n)`
RHS.