Geometric series - Sequences and Series

Geometric series

A geometric series is the sum of a finite number of terms in a geometric sequence. Just like the arithmetic series, we also have geometric series formulas to help us with that.

Basic concepts:
Arithmetic series
Geometric sequences

Related concepts:
Pascal's triangle
Binomial theorem
Introduction to infinite series
Convergence & divergence of geometric series

Lessons

Notes:

• the sum of n terms of a geometric series:
${s_n} = \frac{{{t_1}\;\left( {{r^n} - 1} \right)}}{{r - 1}}$
$=\frac{r \cdot t_{n}-t_{1}}{r-1}$

Geometric series

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Notes:

• the sum of n terms of a geometric series: sn=t1(rn−1)r−1{s_n} = \frac{{{t_1}\;\left( {{r^n} - 1} \right)}}{{r - 1}}s​n​​=​r−1​​t​1​​(r​n​​−1)​​ =r⋅tn−t1r−1=\frac{r \cdot t_{n}-t_{1}}{r-1}=​r−1​​r⋅t​n​​−t​1​​​​

1.

Geometric series formula:sn=t1(rn−1)r−1{s_n} = \frac{{{t_1}\;\left( {{r^n} - 1} \right)}}{{r - 1}}s​n​​=​r−1​​t​1​​(r​n​​−1)​​ Determine the sum of the first twelve terms of the geometric series: 5 – 10 + 20 – 40 + … .

2.

Geometric series formula: sn=r⋅tn−t1r−1s_{n}=\frac{r \cdot t_{n}-t_{1}}{r-1}s​n​​=​r−1​​r⋅t​n​​−t​1​​​​ Determine the sum of the geometric series: 8 + 2 + 12\frac{1}{2}​2​​1​​ + …. + 1512\frac{1}{{512}}​512​​1​​ .

3.

A tennis ball is dropped from the top of a building 15 m high. Each time the ball hits the ground, it bounces back to only 60% of its previous height. What is the total vertical distance the ball has travelled when it hits the ground for the fifth time?

Geometric series

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We have over 1850 practice questions in Algebra for you to master.

Algebra
Arithmetic series

Algebra
Geometric sequences

or

• the sum of n terms of a geometric series:
${s_n} = \frac{{{t_1}\;\left( {{r^n} - 1} \right)}}{{r - 1}}$
$=\frac{r \cdot t_{n}-t_{1}}{r-1}$

Geometric series formula: ${s_n} = \frac{{{t_1}\;\left( {{r^n} - 1} \right)}}{{r - 1}}$
Determine the sum of the first twelve terms of the geometric series: 5 – 10 + 20 – 40 + … .

Geometric series formula: $s_{n}=\frac{r \cdot t_{n}-t_{1}}{r-1}$
Determine the sum of the geometric series: 8 + 2 + $\frac{1}{2}$ + …. + $\frac{1}{{512}}$ .