SOLUTION: If S_17= 629 and a_1= -3 in an arithmetic sequence, find d and a_29 d= a_29=

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Question 1160989: If S_17= 629 and a_1= -3 in an arithmetic sequence, find d and a_29
d=
a_29=
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.

            It is a nice entertainment problem for finding a non-standard solution.
            I will show you the way right now. 


Notice that  the term  a%5B9%5D  is exactly half-way between the terms  a%5B1%5D and  a%5B17%5D.

Therefore,  a%5B9%5D = S%5B19%5D%2F17 = 629%2F17 = 37.


Next, there are 8 gaps between  a%5B1%5D  and  a%5B9%5D;  so, each gap is  %2837+-+%28-3%29%29%2F8 = 40%2F8 = 5.


Thus the common difference of the AP is 5.


Now you know everything about the AP; in particular,  a%5B29%5D = a%5B1%5D + (29-1)*d = -3 + 28*5 = 137.


ANSWER.  d = 5;   a%5B29%5D = 137.

Solved.

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    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Calculating partial sums of arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Advanced problems on arithmetic progressions
    - Interior angles of a polygon and Arithmetic progression
    - Math Olympiad level problem on arithmetic progression
    - Problems on arithmetic progressions solved MENTALLY
    - Mathematical induction and arithmetic progressions
    - Mathematical induction for sequences other than arithmetic or geometric

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
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The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".


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Free of charge online textbook in ALGEBRA-II
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Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The non-standard solution shown by tutor @ikleyn is a good one.... Here is one that is more traditional.

(1) Use the given sum of the first 17 terms and the given first term to find the 17th term.

S%5B17%5D+=+17%28%28a%5B1%5D%2Ba%5B17%5D%29%2F2%29
629+=+17%28%28%28-3%29%2Ba%5B17%5D%29%2F2%29
74+=+-3%2Ba%5B17%5D
a%5B17%5D+=+77

(2) Use the 1st and 17th terms to find the common difference.

a%5B17%5D+=+a%5B1%5D%2B16d
77+=+%28-3%29%2B16d
80+=+16d
d+=+5

The common difference is 5.

(3) Use the first term and the common difference to find the 29th term.

a%5B29%5D+=+a%5B1%5D%2B28d
a%5B29%5D+=+%28-3%29%2B28%285%29+=+%28-3%29%2B140+=+137

The 29th term is 137.