Geometric Series

Calculate sums of geometric series and use formulas. Grade 11 Math 20-1.

Lesson 3.4 of Sequences and Series in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• What is a Geometric Series?. Discover how summing a geometric sequence produces a geometric series. Build intuition by computing partial sums by hand, then learn the two sum formulas and when to use each one.
• Using the Formulas. Apply both geometric series formulas across four worked examples: finding sums when you know the number of terms, when you know the last term, and when the ratio is a fraction or decimal.
• Working Backwards. Flip the formulas around to find missing values. Recover the first term from a known sum, determine how many terms produce a given sum, find the number of terms when only two terms and the total are given, and extract the common ratio from a table of partial sums.
• Applications. Use geometric series to solve real-world problems involving ancestor counting, pendulum decay, variable expressions, and the explosive growth of doubling pennies over a school year.

Inside the lesson — a free preview

Definition

Geometric Series. A geometric series is the sum of the terms of a geometric sequence.

For the geometric sequence 5, -15, 45, -135, , the corresponding geometric series is:

5 + (-15) + 45 + (-135) +

Just as with arithmetic series, S_n denotes the sum of the first n terms, also called a partial sum.

Building Intuition: First Four Partial Sums

Before using a formula, let's see what partial sums look like for a small series.

Example — 5 - 15 + 45 - 135 + where t_1 = 5, r = -3. S_1 = t_1 = 5

S_2 = S_1 + t_2 = 5 + (-15) = -10

S_3 = S_2 + t_3 = -10 + 45 = 35

S_4 = S_3 + t_4 = 35 + (-135) = -100

You could keep doing this by hand, but for S_20 or S_50 you'd want a formula.

The Two Sum Formulas

Formula 1 — When You Know t_1, r, and n. S_n = (t_1(r^n - 1))/(r - 1), r ≠ 1

Use this when you know the first term, common ratio, and number of terms. This is the most common formula.

Formula 2 — When You Know t_1, r, and the Last Term t_n. S_n = (r · t_n - t_1)/(r - 1), r ≠ 1

Use this when you know the first term, common ratio, and the value of the last term being added.

Why r ≠ 1?. If r = 1, every term is just t_1 — no change at all. The series becomes t_1 + t_1 + t_1 + = n × t_1. Both formulas would divide by zero, which is why this case is excluded.

Worked example: a: Find S_7 for 5 - 30 + 180 - 1080 +

We know the first term and can find the ratio, but the last term is unknown — Formula 1 is the right choice.

Example 1a — Find S_7 for 5 - 30 + 180 - 1080 + . Step 1 — Identify what we know:

t_1 = 5

r = (-30)/(5) = -6

n = 7

Step 2 — Plug into Formula 1:

S_n = (t_1(r^n - 1))/(r - 1)

S_7 = (5((-6)^7 - 1))/(-6 - 1)

Step 3 — Evaluate (-6)^7:

(-6)^7 = -279 936 (negative because the exponent is odd)

Step 4 — Substitute and simplify:

S_7 = (5(-279 936 - 1))/(-7) = (5(-279 937))/(-7) = (-1 399 685)/(-7)

Answer: S_7 = 199 955

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