7.4 Sigma notation

Evaluate the following summations. If your answer is undefined, write “Undef”.

(a) [math]\displaystyle{\sum^{4}_{k=0}3^k}[/math]

(b) [math]\displaystyle{\sum^{1}_{k=-3}\frac{1}{k^2}}[/math]

(c) [math]\displaystyle{\sum^{10}_{k=1}(4k-7)}[/math]

(d) [math]\displaystyle{\sum^{10}_{k=1}202}[/math]

(e) [math]\displaystyle{\sum^{100}_{k=1}\frac{1}{k(k+1)}}[/math]

Let [math]S=\displaystyle{\sum^{100}_{k=1}(2k-3)}[/math]. Which of the following values is/are equal to [math]S[/math]?

Simplify [math]6(1^2+3^2+5^2+\cdots+(2n-1)^2)+2n[/math].

If [math]1^3+2^3+3^3+\cdots+2048^3=k[/math], what is the value of [math]2^3+4^3+6^3+\cdots+4096^3[/math]?

Let [math]\{a_n\}[/math] be a sequence having the general form [math]a_n=\displaystyle{\sum^n_{k=1}(4k-7)}[/math]. Evaluate [math]a_{100}-a_{99}[/math].

(Calculator) On 7/15 of the lunar calender, when James walked passed a temple, he noticed that there was a tower built with soda cans. The tower had 20 levels which satisfies
$$x^{\text{th}}=2+5+8+\cdots+(3x-1)$$
For example, therer are [math]2+5+8+11+14=40[/math] cans on the fifth level. How many cans of soda are there in total?