Algebra Examples

$- 5 + - 10 + - 15 + - 20$

Step 1

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

$- 5 - 10 - 15 - 20$

Step 2

This is the formula to find the sum of the first $n$ terms of the sequence. To evaluate it, the values of the first and $n$ th terms must be found.

$S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})$

Step 3

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- 5$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - 5$

Step 4

This is the formula of an arithmetic sequence.

$a_{n} = a_{1} + d (n - 1)$

Step 5

Substitute in the values of $a_{1} = - 5$ and $d = - 5$ .

$a_{n} = - 5 - 5 (n - 1)$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$a_{n} = - 5 - 5 n - 5 \cdot - 1$

Step 6.2

Multiply $- 5$ by $- 1$ .

$a_{n} = - 5 - 5 n + 5$

Step 7

Combine the opposite terms in $- 5 - 5 n + 5$ .

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Step 7.1

Add $- 5$ and $5$ .

$a_{n} = - 5 n + 0$

Step 7.2

Add $- 5 n$ and $0$ .

$a_{n} = - 5 n$

Step 8

Substitute in the value of $n$ to find the $n$ th term.

$a_{4} = - 5 \cdot 4$

Step 9

Multiply $- 5$ by $4$ .

$a_{4} = - 20$

Step 10

Replace the variables with the known values to find $S_{4}$ .

$S_{4} = \frac{4}{2} \cdot (- 5 - 20)$

Step 11

Divide $4$ by $2$ .

$S_{4} = 2 \cdot (- 5 - 20)$

Step 12

Subtract $20$ from $- 5$ .

$S_{4} = 2 \cdot - 25$

Step 13

Multiply $2$ by $- 25$ .

$S_{4} = - 50$