SOLUTION: Question: The arithmetic series 5 + 9 + 13 + ... + tn has a sum of 945. How many terms does the series have? I was just having an issue trying to figure out what formula to use.

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Question 662463: Question: The arithmetic series 5 + 9 + 13 + ... + tn has a sum of 945. How many terms does the series have?
I was just having an issue trying to figure out what formula to use. Any help would be greatly appreciated.

Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
The sum s of an arithmetic series with first term a, n terms, and common difference d is:

s+=+n%28+2a+%2B+%28n+-+1%29d+%29+%2F+2

Applying this to the given series:

945+=+n%28+2+%2A+5+%2B+4%28n+-+1%29+%29+%2F+2
945+=+n%28+10+%2B+4n+-+4+%29+%2F+2

945+=+2n%28+4n+%2B6+%29+%2F+2

945+=+2n%28+2n+%2B3+%29+%2F+2
945+=+cross%282%29n%28+2n+%2B3+%29+%2Fcross%28+2%29

945+=+n%28+2n+%2B3+%29+

945+=+2n%5E2+%2B3n++

2n%5E2+%2B+3n+-+945+=+0...factor...replace 3n with -42n%2B+45n

2n%5E2+-42n%2B+45n+-+945+=+0....group

%282n%5E2+-42n%29%2B+%2845n+-+945%29+=+0

2n%28n+-21%29%2B+45%28n+-+21%29+=+0

%282n+%2B+45%29%28n+-+21%29+=+0

The positive root n+=+21 gives the number of terms.
so, your sequence is:
5, 9, 13,17,21,25,29,33,37,41,45,49,53,57,61,64,69,73,77,81,85
the sum is:

945+=+945



Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
5 + 9 + 13 + ... + tn = 945

We need the sum formula:

Sn = n%2F2[2t1 + (n-1)·d]

To find d, the common difference, we subtract the first term,
t1 = 5 from the second term t2 = 9, and
get 9-5 = 4, and as a check we also subtract the second term,
t2 = 9 from the third term t3 = 12, and
get 13-9 = 4, so the common difference d is 4.  So we substitute
d = 4, Sn = 945, t1 = 5 and solve for n:

Sn = n%2F2[2t1 + (n-1)·d]
945 = n%2F2[2·5 + (n-1)·4]

Multiply both sides by 2 to clear the fraction on the right:

1890 = n[10 + (n-1)·4]

1890 = n[10 + 4(n-1)]

1890 = n[10 + 4n - 4]

1890 = n[4n + 6]

1890 = 4n² + 6n

Divide through by 2

945 = 2n² + 3n

Get 0 on the left

0 = 2n² + 3n - 945

The right side factors as

0 = (n - 21)(2n + 45)

Use the zero-factor property:

    n - 21 = 0;   2n + 45 = 0
         n = 21;       2n = -45
                        n = -45%2F2

Ignore the negative answer.

It has 21 terms.

5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 + 37 + 41 + 45 + 49 + 53 + 57 + 61 + 65 + 69 + 73 + 77 + 81 + 85

Edwin