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Sum of squares and a geometric sum

The sum of the first square numbers is $n(n + 1)(2n + 1)/6$ . It too is proved by induction, with base case $1 = 1$ and a step where you add $(k + 1)^2$ . A geometric sum of powers of 2 is one less than the next power: $1 + 2 + 4 + \dots + 2^{n-1} = 2^n - 1$ . Both formulas can be checked for a small $n$ before you use them for a large one.

1^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}

sum of squares

1 + 2 + \dots + 2^{n-1} = 2^n - 1

powers of 2

Symbols

n

the number of terms

Example

$1^2 + \dots + 10^2 = \dfrac{10\cdot 11\cdot 21}{6} = 385$ . $1 + 2 + 4 + \dots + 2^9 = 2^{10} - 1 = 1023$ .

In the geometric sum the last term is

2^{n-1}

, and the sum is

2^n - 1

Practise proof by induction for free →

← Sum of the first odd numbers · Distance from a point to a plane →

Part of Mathematics R2: Proof by induction.