Answer:
(a) 3, 5, 10, 18, 29, 43
(b) 5, 7, 12, 20, 31, 45
(c) 8, 10, 15, 23, 34, 48
Explanation:
You want the sequences with first differences {2, 5, 8, 11, 14} given that ...
- the first term is 3
- the first 2 terms total 12
- the 5th term is 34.
Cumulative sum
If we start with a first term of 0, we can add the first differences successively to get the corresponding sequence. The sequence of cumulative sums is ...
{0, 2, 7, 15, 26, 40} . . . . . . . "cumsum" sequence
This sequence has a first term of 0 and the given first differences.
(a) First term 3
Adding 3 to the "cumsum" sequence above gives the desired first term and first differences:
3, 5, 10, 18, 29, 43
(b) First 2 terms total 12
The cumsum sequence tells us that the sum of the first two terms of a sequence starting with n will be ...
(n) +(n+2) = 2n+2
We want this value to be 12, so ...
2n +2 = 12
2n = 10 . . . . . . subtract 2
n = 5 . . . . . . . divide by 2
Adding 5 to our "cumsum" sequence gives the sequence with the desired sum of terms:
5, 7, 12, 20, 31, 45 . . . . . . . . . 5+7=12
(c) Fifth term 34
The 5th term of our "cumsum" sequence is 26, so we have to add 8 to make it 34. Adding 8 to the "cumsum" sequence gives the sequence with the desired 5th term:
8, 10, 15, 23, 34, 48
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