## What is Arithmetic Sequence

### Arithmetic Sequences Definition

A list of numbers with a clear pattern is known as an arithmetic sequence. We can tell if a series of numbers is arithmetic by taking any number and subtracting it by the number before it. In an arithmetic sequence, the difference between two numbers stays the same.

The letter $$d$$ stands for the common difference, the constant difference between the numbers in an arithmetic sequence. We have an increasing arithmetic sequence if $$d$$ is positive. We have a decreasing arithmetic sequence if $$d$$ is negative.

### Arithmetic Sequences Formula

We can use the following formula to find different terms for the arithmetic sequence:

$$a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)$$, where:

• $$a_1$$: The first term
• $$a_n$$: The $$n$$th term
• $$n$$: The term poistion
• $$d$$: Common difference

### How to figure out the terms in an arithmetic sequence?

To find any term in the arithmetic sequence, we need to know a common difference, the position of the term we want to find, and a term in the sequence.

### Example

Consider the arithmetic sequence: $$3, \ 7, \ 11, \ 15, \ 19, \ ...$$, Find $$a_{28} \ = \ ?$$

Solution

Before using the arithmetic sequence formula, we need to know the first term, the common difference, and the position of the term we want to find:

• The first term: $$a_1 \ = \ 3$$
• common difference: $$d \ = \ 4$$
• Term's position: $$n \ = \ 28$$

Now, we put these numbers into the formula and find the answer:

$$a_n \ = \ a_1 \ + \ d \ (n \ –\ 1) \ ⇒ \ a_{28} \ = \ 3 \ + \ 4(28 \ - \ 1) \ ⇒ \ a_{28} \ = \ 111$$

### Summing an Arithmetic Series

To summing the terms in an arithmetic sequence, use this formula:

$$\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} (2a \ + \ (n \ - \ 1)d)$$

### Example

Find the sum of the first $$12$$ terms for the sequence: $$1, \ 6, \ 11, \ 16, \ 21, \ ...$$

Solution

• The first term: $$a_1 \ = \ 1$$
• common difference: $$d \ = \ 5$$
• $$n \ = \ 12$$

So:

$$\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} (2a \ + \ (n \ - \ 1)d) \ ⇒$$

$$\sum_{k \ = \ 0}^{11} \ (1 \ + \ 5k) \ = \ \frac{12}{2} (2 \ + \ (11)5) \ = \ 6 \times (57) \ = \ 342$$

### Exercises for Arithmetic Sequences

1) Find the explicit formula: $$5, \ 14, \ 23, \ 32, \ 41, \ ...$$

2) Find the explicit formula: $$-7, \ -2, \ 3, \ 8, \ 13, \ ...$$

3) Find the explicit formula and $$a_{12}$$: $$-22, \ -16, \ -10, \ -4, \ 2, \ ...$$

4) Find the explicit formula and $$a_{21}$$: $$a_{30} \ = \ 72, \ d \ = \ -5$$

5) Find the first $$10$$ terms: $$a_{15} \ = \ 50, \ d \ = \ 4.5$$

6) Find the first $$7$$ terms: $$a_{41} \ = \ 178, \ d \ = \ 4$$

7) Find $$a_{30}$$: $$45, \ , \ 41.8, \ 38.6, \ 35.4, \ ...$$

8) Find $$a_{27}$$: $$a_{41} \ = \ 167, \ d \ = \ 6.4$$

9) Find the explicit formula and the sum of the first five terms: $$a_1 \ = \ 10, \ d \ = \ 3$$

10) Find the explicit formula and the sum of the first $$16$$ terms: $$-55, \ -48, \ -41, \ -34, \ -27, \ ...$$

1) Find the explicit formula: $$5, \ 14, \ 23, \ 32, \ 41, \ ...$$

$$\color{red}{a_1 \ = \ 5, \ d \ = \ 14 \ - \ 5 \ = \ 9, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ 5 \ + \ 9 \ (n \ – \ 1)}$$

2) Find the explicit formula: $$-7, \ -2, \ 3, \ 8, \ 13, \ ...$$

$$\color{red}{a_1 \ = \ -7, \ d \ = \ -2 \ - \ (-7) \ = \ 5, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ -7 \ + \ 5 \ (n \ – \ 1)}$$

3) Find the explicit formula and $$a_{12}$$: $$-22, \ -16, \ -10, \ -4, \ 2, \ ...$$

$$\color{red}{a_1 \ = \ -22, \ d \ = \ -16 \ - \ (-22) \ = \ 6, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ -22 \ + \ 6 \ (n \ – \ 1) \ ⇒ \ a_{12} \ = \ -22 \ + \ 6 \ (12 \ – \ 1) \ = \ 44}$$

4) Find the explicit formula and $$a_{21}$$: $$a_{30} \ = \ 72, \ d \ = \ -5$$

$$\color{red}{a_{30} \ = \ 72, \ d \ = \ -5, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_{30} \ = \ a_1 \ + \ (-5) \ (30 \ – \ 1) \ ⇒ \ a_1 \ = \ a_{30} \ + \ 5(29) \ = \ 72 \ + \ 145 \ = \ 217}$$
$$\color{red}{⇒ \ a_n \ = \ 217 \ - \ 5 \ (n \ – \ 1) \ ⇒ \ a_{21} \ = \ 217 \ - \ 5(20) \ = \ 117}$$

5) Find the first $$10$$ terms: $$a_{15} \ = \ 50, \ d \ = \ 4.5$$

$$\color{red}{a_{15} \ = \ 50, \ d \ = \ 4.5, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_{15} \ = \ a_1 \ + \ 4.5 \ (15 \ – \ 1) \ ⇒ \ a_1 \ = \ 50 \ - \ 4.5(14) \ = \ -13}$$
$$\color{red}{⇒ -13, \ -8.5, \ -4, \ 0.5, \ 5, \ 9.5, \ 14, \ 18.5, \ 23, \ 27.5, \ ...}$$

6) Find the first $$7$$ terms: $$a_{41} \ = \ 178, \ d \ = \ 4$$

$$\color{red}{a_{41} \ = \ 178, \ d \ = \ 4, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_{41} \ = \ a_1 \ + \ 4 \ (41 \ – \ 1) \ ⇒ \ a_1 \ = \ 178 \ - \ 4(40) \ = \ 18}$$
$$\color{red}{⇒ 18, \ 22, \ 26, \ 30, \ 34, \ 38, \ 42, \ ...}$$

7) Find $$a_{30}$$: $$45, \ , \ 41.8, \ 38.6, \ 35.4, \ ...$$

$$\color{red}{a_1 \ = \ 45, \ d \ = \ 41.8 \ - \ 45 \ = \ -3.2, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ 45 \ - \ 3.2 \ (n \ – \ 1) \ ⇒ \ a_{30} \ = \ 45 \ - \ 3.2 \ (30 \ – \ 1) \ = \ -47.8}$$

8) Find $$a_{27}$$: $$a_{41} \ = \ 167, \ d \ = \ 6.4$$

$$\color{red}{a_{41} \ = \ 167, \ d \ = \ 6.4, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_{41} \ = \ a_1 \ + \ 6.4 \ (41 \ – \ 1) \ ⇒ \ a_1 \ = \ 167 \ - \ 6.4(40) \ = \ -98}$$
$$\color{red}{⇒ \ a_n \ = \ -98 \ + \ 6.4 \ (n \ – \ 1) \ ⇒ \ a_{27} \ = \ -98 \ + \ 6.4(26) \ = \ 68.4}$$

9) Find the explicit formula and the sum of the first five terms: $$a_1 \ = \ 10, \ d \ = \ 3$$

$$\color{red}{a_1 \ = \ 10, \ d \ = \ 3, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ 10 \ + \ 3 \ (n \ – \ 1)}$$

$$\color{red}{\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} \ (2a \ + \ (n \ - \ 1)d)}$$

$$\color{red}{⇒ \ \sum_{k \ = \ 0}^{4} \ (10 \ + \ 3k) \ = \ \frac{5}{2} \ (20 \ + \ (4)3) \ = \ \frac{5}{2} \times (32) \ = \ 80}$$

10) Find the explicit formula and the sum of the first $$16$$ terms: $$-55, \ -48, \ -41, \ -34, \ -27, \ ...$$

$$\color{red}{a_1 \ = \ -55, \ d \ = \ -48 \ - \ (-55) \ = \ 7, \ a_n \ = \ a_1 \ + \ d \ (n \ – \ 1)}$$
$$\color{red}{⇒ \ a_n \ = \ -55 \ + \ 7 \ (n \ – \ 1)}$$

$$\color{red}{\sum_{k \ = \ 0}^{n \ - \ 1} \ (a \ + \ kd) \ = \ \frac{n}{2} \ (2a \ + \ (n \ - \ 1)d)}$$

$$\color{red}{⇒ \ \sum_{k \ = \ 0}^{15} \ (-55 \ + \ 7k) \ = \ \frac{16}{2} \ (-110 \ + \ (15)7) \ = \ 8 \times (-5) \ = \ -40}$$

## Arithmetic Sequences Practice Quiz

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