This question considers different methods of summing powers of integers.
The sum of the first \(n\) positive integers, each raised to the power zero, is given by \(S_0(n)=1^0+2^0+\cdots+n^0\), where \(n\in\mathbb{Z}^+\).
Find the value of \(S_0(4)\).[1]
Write down a simplified expression for \(S_0(n)\).[1]
The sum of the first \(n\) positive integers, each raised to the power 1, is given by \(S_1(n)=1^1+2^1+\cdots+n^1\), where \(n\in\mathbb{Z}^+\).
Use the fact that \(S_1(n)\) is the sum of \(n\) terms of an arithmetic sequence to show that \(S_1(n)=\frac{n^2}{2}+\frac n2\).[3]
The sum of the first \(n\) square numbers is given by \(S_2(n)=1^2+2^2+3^2+\cdots+n^2\), where \(n\in\mathbb{Z}^+\).
Find \(S_2(5)\).[1]
The following graph shows three rectangles, where the heights of the rectangles are consecutive square numbers, and the width of each rectangle is \(1\). The graph may be extended in a similar way to include any number of rectangles.
The curve \(y=f(x)\) passes through the top right corner of each rectangle.
Write down an expression for \(f(x)\).[1]
By considering the total area of the first \(n\) rectangles and the area under the curve, show that \(S_2(n)>\frac{n^3}{3}\).[4]
Use a similar method to show that \(S_2(n)<\frac{n^3}{3}+n^2+n\).[4]
Consider the sequence \(u_n=n^3-(n-1)^3\) for \(n\in\mathbb{Z}^+\). Some values of \(u_n\) and \(∑_{r=1}^n u_r\) are shown in the following table.
| \(n\) | \(u_n\) | \(∑_{r=1}^n u_r\) |
|---|
| 1 | 1 | 1 |
| 2 | 7 | 8 |
| 3 | \(a\) | \(b\) |
| 4 | 37 | 64 |
Find the value of \(a\).[1]
Find the value of \(b\).[1]
Suggest an expression for \(∑_{r=1}^n u_r\) in terms of \(n\).[1]
Use the definition of \(u_n\) to prove that your suggestion is true.[3]
Find a quadratic expression for \(u_n\).[2]
Hence, write \(∑_{r=1}^n u_r\) in terms of \(S_2(n)\), \(S_1(n)\) and \(S_0(n)\).[2]
Hence, show that \(S_2(n)=\frac{n^3}{3}+\frac{n^2}{2}+\frac n6\).[3]
Show that the expression for \(S_2(n)\) given in part (g)(iii) is consistent with the inequalities in part (d).[2]