Precalculus Examples

$1 + 3 + 5 + 7 + \dots + 101$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $2$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = 2$

Step 2

Use the formula for an arithmetic sequence $a_{n} = a_{1} + d (n - 1)$ to find the number of terms, $n$ .

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Step 2.1

Substitute the values of the first term, last term, and difference between terms into the formula.

$101 = 1 + 2 (n - 1)$

Step 2.2

Solve for $n$ .

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Step 2.2.1

Rewrite the equation as $1 + 2 (n - 1) = 101$ .

$1 + 2 (n - 1) = 101$

Step 2.2.2

Simplify $1 + 2 (n - 1)$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Apply the distributive property.

$1 + 2 n + 2 \cdot - 1 = 101$

Step 2.2.2.1.2

Multiply $2$ by $- 1$ .

$1 + 2 n - 2 = 101$

Step 2.2.2.2

Subtract $2$ from $1$ .

$2 n - 1 = 101$

Step 2.2.3

Move all terms not containing $n$ to the right side of the equation.

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Step 2.2.3.1

Add $1$ to both sides of the equation.

$2 n = 101 + 1$

Step 2.2.3.2

Add $101$ and $1$ .

$2 n = 102$

Step 2.2.4

Divide each term in $2 n = 102$ by $2$ and simplify.

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Step 2.2.4.1

Divide each term in $2 n = 102$ by $2$ .

$\frac{2 n}{2} = \frac{102}{2}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{2 n}{2} = \frac{102}{2}$

Step 2.2.4.2.1.2

Divide $n$ by $1$ .

$n = \frac{102}{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Divide $102$ by $2$ .

$n = 51$

Step 3

Use the formula for the sum of an arithmetic sequence $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ to find the sum.

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Step 3.1

Substitute the values of the first term, last term, and the number of terms into the sum formula.

$S_{n} = \frac{51}{2} (1 + 101)$

Step 3.2

Simplify.

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Step 3.2.1

Add $1$ and $101$ .

$S_{n} = \frac{51}{2} \cdot 102$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Factor $2$ out of $102$ .

$S_{n} = \frac{51}{2} \cdot (2 (51))$

Step 3.2.2.2

Cancel the common factor.

$S_{n} = \frac{51}{2} \cdot (2 \cdot 51)$

Step 3.2.2.3

Rewrite the expression.

$S_{n} = 51 \cdot 51$

Step 3.2.3

Multiply $51$ by $51$ .

$S_{n} = 2601$