Precalculus Examples

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

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

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding

2

$2$ to the previous term in the sequence gives the next term. In other words,

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

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

Arithmetic Sequence:

d = 2

$d = 2$

Step 2

Use the formula for an arithmetic sequence

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

$a_{n} = a_{1} + d (n - 1)$ to find the number of terms,

n

$n$ .

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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)

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

Step 2.2

Solve for

n

$n$ .

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Step 2.2.1

Rewrite the equation as

1 + 2 (n - 1) = 101

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

1 + 2 (n - 1) = 101

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

Step 2.2.2

Simplify

1 + 2 (n - 1)

$1 + 2 (n - 1)$ .

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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

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

Step 2.2.2.1.2

Multiply

2

$2$ by

- 1

$- 1$ .

1 + 2 n - 2 = 101

$1 + 2 n - 2 = 101$

1 + 2 n - 2 = 101

$1 + 2 n - 2 = 101$

Step 2.2.2.2

Subtract

2

$2$ from

1

$1$ .

2 n - 1 = 101

$2 n - 1 = 101$

2 n - 1 = 101

$2 n - 1 = 101$

Step 2.2.3

Move all terms not containing

n

$n$ to the right side of the equation.

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Step 2.2.3.1

Add

1

$1$ to both sides of the equation.

2 n = 101 + 1

$2 n = 101 + 1$

Step 2.2.3.2

Add

101

$101$ and

1

$1$ .

2 n = 102

$2 n = 102$

2 n = 102

$2 n = 102$

Step 2.2.4

Divide each term in

2 n = 102

$2 n = 102$ by

2

$2$ and simplify.

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Step 2.2.4.1

Divide each term in

2 n = 102

$2 n = 102$ by

2

$2$ .

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

$2$ .

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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

$51$ .

$S_{n} = 2601$