Q.If $\dfrac{1\times3+2\times5+3\times7+\cdots \text{ upto } n \text{ terms}}{1^2+2^2+3^2+\cdots \text{ upto } n \text{ terms}}=\dfrac59$, find the value of $n$.

Beyond a plain A.P. or G.P., many sequences are built from sums or products of simpler patterns, and can be summed using three standard results together with the linearity of summation: the sum of the first $n$ natural numbers is $\sum r=\dfrac{n(n+1)}2$, the sum of their squares is $\sum r^2=\dfrac{n(n+1)(2n+1)}6$, and the sum of their cubes is $\sum r^3=\left[\dfrac{n(n+1)}2\right]^2$. Summation is linear -- $\sum(a_r+b_r)=\sum a_r+\sum b_r$ and $\sum k\,t_r=k\sum t_r$ for a constant $k$, and $\sum 1$ ($n$ times) $=n$ -- so a complicated-looking series is handled by first finding the algebraic expression for its gen …