Associated Topics || Dr. Math Home || Search Dr. Math

### Sum of a Sequence

```
Date: 06/10/99 at 14:25:20
From: Bethany
Subject: Sum of an embedded arithmatic sequence

Our sequence is (3, 4, 6, 9, 13, ..., 499503). We know that there are
1000 terms in the sequence. How would we figure out a formula for the
sum of this sequence? We used Gauss' technique to find the number of
terms by plugging in like this: 3+(n-1)(n-1+1)/2 = 499503.  We got
1000 terms. We know this is correct. How would we find the sum of the
sequence?
```

```
Date: 06/10/99 at 16:45:00
From: Doctor Anthony
Subject: Re: Sum of an embedded arithmatic sequence

Make up a difference table

n =  1     2     3    4      5      6      7 ........
f(n) =  3     4     6    9     13     18     24 ........

1st Diff   1     2     3    4      5      6
2nd Diff      1     1     1     1      1

If the second differences are constant, the nth term will be a

So we assume    f(n) =  an^2 + bn + c

n=1                 a +  b +  c  =  3
n=2                4a + 2b +  c  =  4
n=3                9a + 3b +  c  =  6

So we have 3 equations with 3 unknowns a, b, and c.

The solutions are   a = 1/2,  b = -1/2,  c = 3

And therefore   f(n) = (1/2)n^2 - (1/2)n + 3

Check with n = 5: f(n) = 12.5 - 2.5 + 3 = 13, which checks.

n = 1000  so we get: f(n) = 500000 - 500 + 3 = 499503, which also
checks.

So we must sum the following

1000
SUM[n^2/2 - n/2 + 3]
n=1

and using the standard formulae for SUM(n^2) and SUM(n) we get

(1/2)n(n+1)(2n+1)/6 - (1/2)n(n+1)/2 + 3n

n(n+1)[2n+1 - 3]
= ----------------  + 3n
12

n(n+1)(2n-2)
= ------------ + 3n
12

n(n+1)(n-1)           n(n^2 - 1) + 18n
= ----------- + 3n   =  ----------------
6                       6

Check when n = 3. The sum of the first 3 terms should be 13

3 x (9 - 1) + 54
----------------  =  13  so our formula is correct.
6

Finally, put n = 1000 into the formula

1000(1000^2 - 1) + 18000
------------------------  =  166669500
6

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search