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College Algebra

Series and Sigma Notation

Lesson

A seriesis the sum of a sequence’s terms. When the sequence is arithmetic or geometric, there’s a closed formula — you don’t have to add term by term.

Sigma notation compresses a sum into one line:

∑i=1nf(i)=f(1)+f(2)+⋯+f(n)\sum_{i=1}^{n} f(i) = f(1) + f(2) + \cdots + f(n)

The ∑\sum says “sum,” the index ii starts at the bottom value and runs up to the top value, and you plug each ii into f(i)f(i) and add the results.

The standard formulas:

Arithmetic series: Sn=n2 (a1+an)\text{Arithmetic series: } S_n = \frac{n}{2}\,(a_1 + a_n)
Geometric series: Sn=a1⋅1−rn1−r\text{Geometric series: } S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

Infinite geometric series, when ∣r∣<1|r| < 1:

S∞=a11−rS_\infty = \frac{a_1}{1 - r}

Worked example 1 — sigma

∑i=14i2\sum_{i=1}^{4} i^2

Plug each ii from 1 to 4 into i2i^2 and add:

1+4+9+16=301 + 4 + 9 + 16 = 30

Worked example 2 — geometric series

Find the sum of the first 4 terms of the geometric sequence with a1=2, r=3a_1 = 2,\ r = 3.

S4=2⋅1−341−3=2⋅−80−2=80S_4 = 2 \cdot \frac{1 - 3^4}{1 - 3} = 2 \cdot \frac{-80}{-2} = 80

Worked example 3 — infinite geometric

With a1=4a_1 = 4 and r=12r = \tfrac{1}{2} (so ∣r∣<1|r| < 1):

S∞=41−12=412=8S_\infty = \frac{4}{1 - \tfrac{1}{2}} = \frac{4}{\tfrac{1}{2}} = 8

How to type your answer

Type a single number — the sum. Use a slash for fractions. Examples: 55, 820, 27/2, 5050.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

1+2+3+⋯+10=?1 + 2 + 3 + \cdots + 10 = ?

Problem 2

2+4+6+8+10=?2 + 4 + 6 + 8 + 10 = ?

Problem 3

1+3+5+7+9=?1 + 3 + 5 + 7 + 9 = ?

Problem 4

5+10+15+20=?5 + 10 + 15 + 20 = ?

Practice

Standard problems matching the lesson.

Problem 5

∑i=15i=?\sum_{i=1}^{5} i = ?

Problem 6

∑i=14i2=?\sum_{i=1}^{4} i^2 = ?

Problem 7

∑i=132i=?\sum_{i=1}^{3} 2i = ?

Problem 8

∑i=15(i+3)=?\sum_{i=1}^{5} (i + 3) = ?

Problem 9

Arithmetic, a1=2, d=3. S10=?\text{Arithmetic, } a_1 = 2,\ d = 3.\ S_{10} = ?

Problem 10

Arithmetic, a1=5, d=4. S8=?\text{Arithmetic, } a_1 = 5,\ d = 4.\ S_8 = ?

Problem 11

Geometric, a1=2, r=3. S4=?\text{Geometric, } a_1 = 2,\ r = 3.\ S_4 = ?

Problem 12

Geometric, a1=1, r=2. S5=?\text{Geometric, } a_1 = 1,\ r = 2.\ S_5 = ?

Problem 13

Sum of first 20 terms of 3, 7, 11, 15, …\text{Sum of first 20 terms of } 3,\,7,\,11,\,15,\,\ldots

Problem 14

5+10+20+40+80=?5 + 10 + 20 + 40 + 80 = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

∑i=110i=?\sum_{i=1}^{10} i = ?

Problem 16

∑i=16i2=?\sum_{i=1}^{6} i^2 = ?

Problem 17

∑i=15(2i−1)=?\sum_{i=1}^{5} (2i - 1) = ?

Problem 18

∑i=143⋅2 i−1=?\sum_{i=1}^{4} 3 \cdot 2^{\,i-1} = ?

Problem 19

∑i=15(3i+2)=?\sum_{i=1}^{5} (3i + 2) = ?

Problem 20

Infinite geometric: a1=4, r=12. S∞=?\text{Infinite geometric: } a_1 = 4,\ r = \tfrac{1}{2}.\ S_\infty = ?

Problem 21

Infinite geometric: a1=9, r=13. S∞=?\text{Infinite geometric: } a_1 = 9,\ r = \tfrac{1}{3}.\ S_\infty = ?

Problem 22

∑i=1100i=?\sum_{i=1}^{100} i = ?