\displaystyle \textbf{1. Standard Summation Formulae}
\displaystyle \text{For any }n\in N\text{, we have:}
\displaystyle \text{(i) }\sum_{k=1}^{n}k=\frac{n(n+1)}{2}
\displaystyle \text{(ii) }\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}
\displaystyle \text{(iii) }\sum_{k=1}^{n}k^3=\left(\frac{n(n+1)}{2}\right)^2
\displaystyle \text{(iv) }\sum_{k=1}^{n}k^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}
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\displaystyle \textbf{2. Series with Successive Differences in A.P.}
\displaystyle \text{If the differences }
\displaystyle a_2-a_1,\;a_3-a_2,\;a_4-a_3,\ldots
\displaystyle \text{are in an A.P., then the nth term is given by}
\displaystyle a_n=an^2+bn+c
\displaystyle \text{where }a,b,c\text{ are constants.}
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\displaystyle \textbf{3. Series with Successive Differences in G.P.}
\displaystyle \text{If the differences }
\displaystyle a_2-a_1,\;a_3-a_2,\;a_4-a_3,\ldots
\displaystyle \text{are in a G.P. with common ratio }r,\text{ then the nth term is given by}
\displaystyle a_n=ar^{\,n-1}+bn+c
\displaystyle \text{where }a,b,c\text{ are constants.}
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\displaystyle \textbf{4. Determination of Constants}
\displaystyle \text{To determine the constants }a,b,c\text{, put }n=1,2,3\text{ and equate them with the values of}
\displaystyle \text{the corresponding terms of the given series.}
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