Suppose that a sequence {am } n> y has constant second-differences. Then the sequence itself is not arithmetic, but its

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Suppose that a sequence {am } n> y has constant second-differences. Then the sequence itself is not arithmetic, but its sequence of
first-differences IS an arithmetic sequence. Call this sequence of first differences {dm }n> 1: Then the original sequence can be
constructed as follows:
• Q1 is given
• a2 = a1 + d1 az
= a1 + di + d2, a1 = a1 + di + d2 + d3, and so on
• the general term may be expressed as an = ay + de + d2 + d3 + •
.. + dr -2 + dm-1
To find a closed formula for Qm, all we need to do is first find a closed formula for the sum
Sn-1 = de + d2 + de + . . . + dn-2 + dn- yin terms of n. Once this is accomplished, the closed formula for our original
sequence will simply be an = a1 + Sn-1.
Suppose that the first term of the original sequence is Q1 =4. Suppose further that the arithmetic sequence of first differences starts
at 7 and increases by 5 from each term to the next.
Use the Reverse-and-Add technique to determine a formula for Sn-1, the sum of the first-differences as described above, in terms of
M. Finally, use the formula for the sum of the differences to determine a general formula for On-
What is the value of An When n =129?