Theory of Intelligence
11TH•Mathematics•Sequences And Series•6.7 Geometric Series

6.7 Geometric Series

Sum of a geometric series and worked examples.

timer2 min read•forumLesson Discussion

6.7 Geometric Series

The sum of terms of a geometric sequence is a geometric series.

Sn=a1+a1r+⋯+a1rn−1S_n = a_1 + a_1 r + \cdots + a_1 r^{n-1}

Multiply by rr and subtract:

Sn=a1(1−rn)1−r,r≠1S_n = \dfrac{a_1(1 - r^n)}{1 - r}, \quad r \neq 1

If r=1r = 1, then Sn=na1S_n = na_1.

Example 18

If an=(−3)(25)na_n = (-3)\bigl(\dfrac{2}{5}\bigr)^n, rewrite as an=(−65)(25)n−1a_n = \bigl(-\dfrac{6}{5}\bigr)\bigl(\dfrac{2}{5}\bigr)^{n-1}. Then a1=−65a_1 = -\dfrac{6}{5}, r=25r = \dfrac{2}{5}, and

Sn=(−2)[1−(25)n]S_n = (-2)\left[1 - \left(\dfrac{2}{5}\right)^n\right]

Discuss theoretical concepts, mathematical formulas, and architectural foundations.

Supports inline code with `backticks`
No comments yet. Be the first to start the discussion!
arrow_backPrevious Lesson
6.6 Geometric Mean
Next Lessonarrow_forward
6.8 Arithmetico-Geometric Progression