If x1,x2,…,xn are in H.P, then ∑r=1n−1 xrxr+1 is equal to

Clearly 1x1,1x2,−−−1xn will be in A.P ⇒1x2−1x1=1x3−1x2=−−−=1xr+1−1xr=−−⇒1xn−1xn−1=λ(say)⇒∑r=1n−1 xrxr+1=1λ∑r=1n−1 xr−xr+1=1λx1−xn Now __,1xn=1x1+(n−1)λ⇒x1−xnx1xn=(n−1)λ=∑r=1n−1 xrxr+1=(n−1)x1xn