SOLUTION: The first difference of a sequence is the arithmetic sequence 3, 5, 7, 9, 11, ...... Find the first six terms of the original sequence in each of the following cases. A. The fir

Question 1092371: The first difference of a sequence is the arithmetic sequence 3, 5, 7, 9, 11, ...... Find the first six terms of the original sequence in each of the following cases.
A. The first term in the original sequence is 3.
B. The sum of the first two terms in the original sequence is 11.
C. The fifth term in the original is 40.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

This should be pretty straightforward, if you understand what the first differences of a sequence mean. The first differences being 3, 5, 7, 9, 11, ... means
3 is the difference between the 1st and 2nd terms;
5 is the difference between the 2nd and 3rd terms;
7 is the difference between the 3rd and 4th terms;
9 is the difference between the 4th and 5th terms; and
11 is the difference between the 5th and 6th terms

So....
A. If the first term of the sequence is 3, then the 2nd is 3+3 = 6; the 3rd is 6+5 = 11; the 4th is 11+7 = 18; the 5th is 18+9 = 27, and the 6th is 27+11 = 38.

B. If the sum of the first two terms is 11, and we know that the difference between the first two terms is 3, then we know the first two terms are 4 and 7. To get the 3rd through 6th terms, simply add the given differences between terms, as demonstrated in part A.

C. Given that the 5th term of the sequence is 40, you can find the 6th term by adding the known difference between the 5th and 6th terms; and you can find the 4th, 3rd, 2nd, and 1st terms by working backwards, subtracting the known differences between terms.

Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
To wider your horizon:
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If the first difference of a sequence is an arithmetic sequence, then the original sequence is a quadratic sequence.

See the lessons
    -Free fall and arithmetic progressions
    -Uniformly accelerated motions and arithmetic progressions
    -Increments of a quadratic function form an arithmetic progression
in this site.

There is a bunch of lessons on arithmetic progressions in this site:
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Mathematical induction and arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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