If the sum of n terms of a string is 5n^2+6n find the form of the general term.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the sum of n terms of a string given as 5n^2+6n

Now 5n^2+6n = 5n^2 + 5n + n

=> 5n(n+1) + n

=> 10n(n+1)/2 +n

Now 10*n(+1)/2 is the sum of n terms of the form n multiplied by 10 and n is the sum of n terms equal to 1.

Therefore the general term has the form 10n +1.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll note the sum of n terms as Sn.

Sn = a1 + a2 + ... + an

From enunciation, we know that:

Sn = 5n^2 + 6n

a1 + a2 + ... + an = 5n^2 + 6n

an is the general term of the string.

We notice that we can calculate an subtracting a1 + a2 + ... + a(n-1) both sides:

Sn - [a1 + a2 + ... + a(n-1)] = an

But the sum of n-1 terms is S(n-1)

an = Sn - S(n-1)

Since

Sn = 5n^2 + 6n => S(n-1) = 5(n-1)^2 + 6(n-1)

We'll expand the square:

S(n-1) = 5n^2 - 10n + 5 + 6n - 6

We'll combine like terms:

S(n-1) = 5n^2 - 4n - 1

an = 5n^2 + 6n - 5n^2 + 4n + 1

We'll eliminate and combine like terms:

an = 10n + 1

The form of the general term an is:

an = 10n + 1

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