Sequences and Series

Geometric Series Sum

Sum the first n terms of a geometric sequence using S_n = a_1 · (r^n - 1) / (r - 1) when r ≠ 1.

Example

Use the finite geometric series formula to find a sum.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

a1=2r=3n=5a_{ 1 }= 2 \quad r= 3 \quad n= 5

Step 2 — Sum formula

Use the finite geometric series formula.

Sn=a1rn−1r−1S_n=a_{ 1 }\frac{r^n- 1 }{r- 1 }

Step 3 — Substitute

Substitute first term 2, ratio 3, and n = 5.

S5=2⋅35−13−1S_{ 5 }= \hl{2} \cdot\frac{ \hl{3} ^{ \hl{5} }- 1 }{ \hl{3} - 1 }

Step 4 — Power

Compute the power: r to the n is 243.

rn=243r^n= \hl{243}

Step 5 — Result

The finite geometric sum is 242.

S5=242S_{ 5 }= \hl{242}
geometric-series-sum The sum of the first n terms of a geometric sequence: S_n = a_1 · (r^n - 1) / (r - 1) Steps: 1. Compute r^n (the nth power of the ratio). 2. Substitute into the formula and simplify. 3. The denominator r - 1 cancels common factors.