How do you find a formula for the sum n

Priscilla Johnston

Priscilla Johnston

Answered question

2021-12-13

How do you find a formula for the sum n terms i=1n(1+in)(2n) and then find the limit as n

Answer & Explanation

Ben Owens

Ben Owens

Beginner2021-12-14Added 27 answers

i=1n(1+in)(2n)=3n+1n
limni=1n(1+in)(2n)=3
Explanation
Let Sn=i=1n(1+in)(2n)
Sn=i=1n(2n+2in2)
Sn=2ni=1n(1)+2in2i=1n(i)
And using the standard results: i=1nr=12n(n+1)
We have:
Sn=2n(n)=2n212n(n+1)
Sn=2+n+1n
Sn=(2n)+(n+1)n
Sn=3n+1n
Now we examine the behaviour of Sn as n
We have:
Sn=3n+1n
Sn=3+1n
limnSn=limn{3+1n}
limnSn=limn(3)limn(1n)
And as 1n0 as n we have;
limn
Gerald Lopez

Gerald Lopez

Beginner2021-12-15Added 29 answers

Thanks!

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?