# 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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### 2 Answers

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**

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.**