Question

Given the string (an),n>=1 and the sum a1+a2+a3+...+an=(5n^2+6n), what are an and a1?

Accepted Answer

We are given that a1 + a2 + a3 + ... + an = (5n^2+6n).
Sn = a1 + a2 + a3 + ... + an = (5n^2+6n).
Sn+1 = a1 + a2 + a3 + ... + an + an+1 = (5(n+1)^2+6(n+1)).
Sn+1 - Sn
=> a1 + a2 + a3 + ... + an + an+1 - a1 + a2 + a3 + ... + an = (5(n+1)^2+6(n+1)) - (5n^2+6n)
=> an+1 = 5n^2 + 5 + 10n + 6n + 6 - 5n^2 - 6n
=> an+1 = 5 + 10n + 6
=> an+1 = 10n + 11
=> an = 10(n-1) + 11
=> an = 10n - 10 + 11
=> an = 10n + 1
a1 = 10n + 1 = 11
The required value of an = 10n + 1 and a1 = 11

Answer

If the sum a1 + ... + an = 5n^2+6n, then the sum:
a1 + ... + a(n-1) = 5(n-1)^2 + 6(n-1)
We'll determine an:
a1 + ... + an - a1 - ... - a(n-1) = 5n^2 + 6n - 5(n-1)^2 - 6(n-1)
We'll eliminate like terms and we'll get:
an = 5n^2 + 6n - 5(n-1)^2 - 6(n-1)
We'll raise to square and we'll combine like terms from the right side:
an = 5n^2 + 6n - 5n^2 + 10n - 5 - 6n + 6
an = 10n + 1
We can determine any other term of the string, replacing n by any natural value.
For n=1 => a1 = 10+1
a1 = 11
The requested terms are: a1 = 11 and an = 10n + 1.

algebra1

Given the string (an),n>=1 and the sum a1+a2+a3+...+an=(5n^2+6n), what are an and a1?

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