Training Sequences & Series Arithmetic & Geometric Sequences
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Arithmetic & Geometric Sequences

22 min Sequences & Series
Arithmetic sequences have a constant common difference d between consecutive terms: aₙ = a₁ + (n − 1)d. Geometric sequences have a constant common ratio r: aₙ = a₁ · rⁿ⁻¹. Arithmetic sequences model linear growth (salary raises, equally spaced data), while geometric sequences model exponential growth or decay (compound interest, radioactive decay, population dynamics). Recognizing which type applies and extracting d or r from given terms are essential skills. The explicit formula lets you find any term directly, while the recursive form aₙ₊₁ = aₙ + d (or aₙ₊₁ = r · aₙ) is useful for programming and induction proofs.

Arithmetic & Geometric Sequences

Arithmetic Sequence

A sequence with a constant difference $d$ between consecutive terms:

$$a_n = a_1 + (n-1)d$$

Geometric Sequence

A sequence with a constant ratio $r$ between consecutive terms:

$$a_n = a_1 \cdot r^{n-1}$$

Example 1

Find the 20th term of the arithmetic sequence $3, 7, 11, 15, \ldots$

  1. $a_1 = 3$, $d = 4$.
  2. $$a_{20} = 3 + 19(4) = 3 + 76 = 79$$
Example 2

Find the 8th term of the geometric sequence $2, 6, 18, 54, \ldots$

  1. $a_1 = 2$, $r = 3$.
  2. $$a_8 = 2 \cdot 3^7 = 2 \cdot 2187 = 4374$$
Example 3

Is $5, 10, 20, 40, \ldots$ arithmetic or geometric?

  1. $10/5 = 2$, $20/10 = 2$: constant ratio → geometric with $r = 2$.
Example 4

The 5th term of an arithmetic sequence is 23 and $d = 4$. Find $a_1$.

  1. $a_5 = a_1 + 4d \Rightarrow 23 = a_1 + 16 \Rightarrow a_1 = 7$.

Practice Problems

1. Find $a_{15}$: arithmetic, $a_1 = 2$, $d = 5$.
2. Find $a_{10}$: geometric, $a_1 = 3$, $r = 2$.
3. Is $1, 4, 9, 16, \ldots$ arithmetic? Geometric?
4. Find $d$: $a_1 = 10$, $a_6 = 35$.
5. Find $r$: $a_1 = 4$, $a_4 = 108$.
6. Write the first 5 terms: $a_n = 3n - 1$.
7. Write the first 5 terms: $a_n = 2^n$.
8. Find $a_{12}$: $a_1 = 100$, $d = -7$.
9. Geometric: $a_1 = 1000$, $r = 0.5$. Find $a_6$.
10. Arithmetic: $a_3 = 14$, $a_7 = 30$. Find $d$ and $a_1$.
11. $a_n = (-1)^n \cdot n$. List the first 4 terms.
12. Geometric: $a_2 = 12$, $r = 3$. Find $a_1$ and $a_5$.
Show Answer Key

1. $2 + 14(5) = 72$

2. $3 \cdot 2^9 = 1536$

3. Neither (differences and ratios are not constant)

4. $35 = 10 + 5d \Rightarrow d = 5$

5. $108 = 4r^3 \Rightarrow r = 3$

6. $2, 5, 8, 11, 14$

7. $2, 4, 8, 16, 32$

8. $100 + 11(-7) = 23$

9. $1000(0.5)^5 = 31.25$

10. $d = 4$, $a_1 = 6$

11. $-1, 2, -3, 4$

12. $a_1 = 4$, $a_5 = 324$

📈 Arithmetic & Geometric Sequence
Sequence
aₙ
Sum Sₙ
Summation Notation & Partial Sums