Precalculus Examples

Find the Sum of the Series 1+3+5+7+...+101
1+3+5+7++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, an=a1+d(n1).
Arithmetic Sequence: d=2
Step 2
Use the formula for an arithmetic sequence an=a1+d(n1) 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(n1)
Step 2.2
Solve for n.
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Step 2.2.1
Rewrite the equation as 1+2(n1)=101.
1+2(n1)=101
Step 2.2.2
Simplify 1+2(n1).
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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+2n+21=101
Step 2.2.2.1.2
Multiply 2 by 1.
1+2n2=101
1+2n2=101
Step 2.2.2.2
Subtract 2 from 1.
2n1=101
2n1=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.
2n=101+1
Step 2.2.3.2
Add 101 and 1.
2n=102
2n=102
Step 2.2.4
Divide each term in 2n=102 by 2 and simplify.
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Step 2.2.4.1
Divide each term in 2n=102 by 2.
2n2=1022
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.
2n2=1022
Step 2.2.4.2.1.2
Divide n by 1.
n=1022
n=1022
n=1022
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
n=51
n=51
n=51
n=51
Step 3
Use the formula for the sum of an arithmetic sequence Sn=n2(a1+an) 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.
Sn=512(1+101)
Step 3.2
Simplify.
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Step 3.2.1
Add 1 and 101.
Sn=512102
Step 3.2.2
Cancel the common factor of 2.
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Step 3.2.2.1
Factor 2 out of 102.
Sn=512(2(51))
Step 3.2.2.2
Cancel the common factor.
Sn=512(251)
Step 3.2.2.3
Rewrite the expression.
Sn=5151
Sn=5151
Step 3.2.3
Multiply 51 by 51.
Sn=2601
Sn=2601
Sn=2601
 x2  12  π  xdx