Question

Given series:

$n$

$Sn = Σ $ $((k+1)^2 - k^2)$

$k=1$

which of the following statements are true?

1. it is a telescoping series
2. it sums to n(n+1)
3. it sums to n(n+2)
4. it sums to n(n+1) - n
5. it sums to the same value as:

$n$

$Σ$ $(2k + 1)$

 $k = 1$

6) it sums to the same value as:

$n + 1$

$Σ$ $(2k)$

$k = 2$

Answer 1

1.false

A telescoping series is a series whose general term can be written as the difference of two consecutive terms of a sequence:

$t_n=a_n-a_{n+1}$

2.false

$S_n=\sum((k+1)^2-k^2)=(2^2-1)+(3^2-2^2)+(4^2-3^2)+...+((n+1)^2-n^2)=$

$=(n+1)^2-1=n^2+2n=n(n+2)$

3.true

4.false

5.true

$S_n=\sum((k+1)^2-k^2)=\sum(k^2+2k+1-k^2)=\sum(2k+1)$

6.false

$\displaystyle{\sum_2^{n+1}2k}=4+8+10+...+2(n+1)=n(4+2n+2)/2=n(n+3)$