Training Sequences & Series Summation Notation & Partial Sums
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Summation Notation & Partial Sums

24 min Sequences & Series

Summation Notation & Partial Sums

Sigma Notation

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$$

Arithmetic Series

$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$

Geometric Series

$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$$

Useful Summation Formulas

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2} \qquad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

Example 1

Find $\sum_{k=1}^{100} k$.

$$\frac{100 \cdot 101}{2} = 5050$$

Example 2

Find the sum of the first 10 terms of the arithmetic sequence $5, 8, 11, \ldots$

$a_1 = 5$, $d = 3$, $a_{10} = 5 + 27 = 32$.

$$S_{10} = \frac{10}{2}(5 + 32) = 5 \cdot 37 = 185$$

Example 3

Find the sum of the first 6 terms: $3, 6, 12, 24, \ldots$

$a_1 = 3$, $r = 2$.

$$S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2} = 3 \cdot \frac{-63}{-1} = 189$$

Practice Problems

1. Compute $\sum_{k=1}^{5} k^2$.
2. Arithmetic series: $a_1 = 4$, $d = 3$, $n = 20$. Find $S_{20}$.
3. Geometric series: $a_1 = 1$, $r = 3$, $n = 5$. Find $S_5$.
4. Compute $\sum_{k=1}^{50} k$.
5. $\sum_{k=0}^{4} 2^k$.
6. Arithmetic: $2 + 5 + 8 + \cdots + 29$. How many terms? Sum?
7. Geometric: $a_1 = 100$, $r = 0.5$, $n = 8$. Find $S_8$.
8. Write $3 + 6 + 9 + 12 + 15$ in sigma notation.
9. $\sum_{k=1}^{4} (2k+1)$.
10. Find $\sum_{k=1}^{10} k^2$ using the formula.
11. Geometric: $5 + 5 + 5 + 5 + 5$ ($r = 1$). Sum?
12. Arithmetic: $a_1 = 50$, $d = -4$, $n = 15$. Find $S_{15}$.
Show Answer Key

1. $1 + 4 + 9 + 16 + 25 = 55$

2. $S_{20} = \frac{20}{2}[8 + 57] = 10 \cdot 65 = 650$... wait: $a_{20} = 4 + 19(3) = 61$. $S_{20} = 10(4 + 61) = 650$

3. $1 \cdot \frac{1-243}{1-3} = \frac{-242}{-2} = 121$

4. $\frac{50 \cdot 51}{2} = 1275$

5. $1 + 2 + 4 + 8 + 16 = 31$

6. $29 = 2 + (n-1)3 \Rightarrow n = 10$. $S_{10} = 5(2 + 29) = 155$

7. $100 \cdot \frac{1 - (0.5)^8}{0.5} = 200(1 - 1/256) \approx 199.22$

8. $\sum_{k=1}^{5} 3k$

9. $3 + 5 + 7 + 9 = 24$

10. $\frac{10 \cdot 11 \cdot 21}{6} = 385$

11. $25$ (simply $5 \times 5$)

12. $a_{15} = 50 - 56 = -6$. $S_{15} = \frac{15}{2}(50 + (-6)) = 330$

Arithmetic & Geometric Sequences Infinite Series & Convergence