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### Course: Calculus, all content (2017 edition)>Unit 7

Lesson 5: Partial sums

# Partial sums: term value from partial sum

The partial sum of a sequence gives us the sum of the first n terms in the sequence. If we know the formula for the partial sums of a sequence, we can find the value of any term in the sequence.

## Want to join the conversation?

• Why are we substracting S sub 6. Why can´t we just plug in 7 for S sub n?
• Unfortunately, S_7 would be the sum of the first 7 terms. Normally the S_n formula for a partial sum means... a sum of those first n terms! If you just want ONE single term though, the partial sum formula isn't going to help directly.

But, if you subtract the terms prior to that term, then you get the term alone! For example:

S_4 = a1 + a2 + a3 + 4
S_3 = a1 + a2 + a3
S_4 - S_3 = a1 - a1 + a2 - a2 + a3 - a3 + a4
OR S_4 - S_3 = 0 + 0 + 0 + a4 = a4!

That's the gist of it. If you want the nth term in a series, then you take the nth partial sum, and subtract the n-1 partial sum from it, this gives you simply the nth TERM (not the partial sum, but a term you would add toward the sum).
• Why can't we just come up with a rule for a(sub n) and then input 7 into our formula? I tried doing it that way but I ended up with a completely different answer, can you explain why we can't use this method?
• In fact, we can come up with a rule for a_n in this example!

All you have to do is to simplify the following expression which corresponds to S(n) - S(n-1):

(n²+1)/(n+1) - ((n-1)²+1)/((n-1)+1)

If you do the steps correctly, you'll end up with this expression:

a_n = (n²+n-2)/(n²+n).

If you try to evaluate a_7, you will indeed get 27/28 as a result, as expected.

Have a nice day!