Finding the sum of different types of series:
- The sum of a finite arithmetic progression:
[
S_n = \frac{n}{2} (a_1 + a_n)
]
where ( S_n ) is the sum of the first ( n ) terms, ( a_1 ) is the first term, ( a_n ) is the last term, ( n ) is the number of terms. - Sum of an infinite arithmetic progression:
An infinite arithmetic progression does not have a finite sum unless the difference is zero. - The sum of a finite geometric progression:
[
S_n = a_1 \frac{1 – r^n}{1 – r} \quad (r \neq 1)
]
where ( S_n ) is the sum of the first ( n ) terms, ( a_1 ) is the first term, ( r ) is the denominator of the progression. - The sum of an infinite geometric progression:
[
S = \frac{a_1}{1 – r} \quad (|r| < 1)
]
where ( S ) is the sum, ( a_1 ) is the first term, ( r ) is the denominator of the progression. - The sum of a series of natural numbers:
[
S_n = \frac{n(n + 1)}{2}
]
where ( S_n ) is the sum of the first ( n ) natural numbers. - Sum of the squares of the first ( n ) natural numbers:
[
S_n = \frac{n(n + 1)(2n + 1)}{6}
] - Sum of cubes of first ( n ) natural numbers:
[
S_n = \left( \frac{n(n + 1)}{2} \right)^2
]