Summation Formulas
Before going to learn summation formulas, first, we will recall the meaning of summation. Summation (or) sum is the sum of consecutive terms of a sequence. To write the sum of more terms, say n terms, of a sequence {a\(_n\)}, we use the summation notation instead of writing the whole sum manually. i.e., \(a_1+a_2+...+a_n= \sum_{i=1}^{n} a_{i}\). Let us learn the summation formulas and their applications using a few solved examples.
What are Summation Formulas?
The summation formulas are used to calcu;ate the sum of the sequence. There are various types of sequences and hence there are various types of sequence summation formulas. One of the summation formulas is,
\( \sum_{i=1}^{n} 1 \) = 1 + 1 + ... + 1 (n times) = n
Apart from this, we have other summation formulas to find:

the sum of first n natural numbers, which is \(\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\)

the sum of squares of first n natural numbers, which is \(\sum_{i=1}^{n} i^{2}=\frac{n(n+1)(2 n+1)}{6}\)

the sum of cubes of first n natural numbers, which is \(\sum_{i=1}^{n} i^{3}=\frac{n^{2}(n+1)^{2}}{4}\)

the sum of fourth powers of first n natural numbers, which is \(\sum_{i=1}^{n} i^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n1\right)\)
Solved Examples Using Summation Formulas

Example 1: Find the value of \(\sum_{i=1}^{n} (32i)\) using the summation formulas.
Solution:
To find: The given sum using the summation formulas.
\( \begin{align} \sum_{i=1}^{n} (3 2i)&= 3 \sum_{i=1}^{n} 1  2 \sum_{i=1}^{n} i\\[0.2cm] &= 3 n  2 \left( \dfrac{n(n+1)}{2} \right)\\[0.2cm] &= \dfrac{6n 2n^22n}{2}\\[0.2cm] &= \dfrac{4n2n^2}{2}\\[0.2cm] &= 2nn^2 \end{align}\)
Answer: \(\sum_{i=1}^{n} (3 2i) =2nn^2\).

Example 2:
Find the value of the summation \(\sum_{k=1}^{150}(k3)^{2}\) using the summation formulas.
Solution:
To find: The given sum using the summation formulas.
\( \begin{align} \sum_{k=1}^{150}(k3)^{2} &= \sum_{k=1}^{150} (k^2 6k+9)\\[0.2cm] &= \sum_{k=1}^{150} k^2  6 \sum_{k=1}^{150} k + 9 \sum_{k=1}^{150} 1 \\[0.2cm] &= \dfrac{150(150+1)(2(150)+1}{6} 6 \cdot \dfrac{150(150+1)}{2} + 9 (150)\\[0.2cm] &= 1136275 67950 + 1350\\[0.2cm] &=1069675 \end{align}\)
Answer: \( \sum_{k=1}^{150}(k3)^{2}=1,069,675\)