## Question 6:

### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $n=1$: LHS $= 1^3 = 1$, RHS $= \frac{1}{4} \times 1^2 \times 2^2 = 1$ | B1 | Shows **both** LHS $= 1$ **and** RHS $= 1$ |
| Assume true for $n = k$; when $n = k+1$: $\sum_{r=1}^{k+1} r^3 = \frac{1}{4}k^2(k+1)^2 + (k+1)^3$ | M1 | Adds $(k+1)^3$ to the given result |
| $= \frac{1}{4}(k+1)^2[k^2 + 4(k+1)]$ | dM1 | Attempt to factorise out $\frac{1}{4}(k+1)^2$ |
| | A1 | Correct expression with $\frac{1}{4}(k+1)^2$ factored out |
| $= \frac{1}{4}(k+1)^2(k+2)^2$ | A1cso | Fully complete proof with no errors and comment. Must see: true for $n=1$, assumption true for $n=k$, said true for $n=k+1$, therefore true for all $n$ |

### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum(r^3 - 2) = \sum r^3 - \sum 2$ | M1 | Attempt two sums ($\sum r^3 - \sum 2n$ is M0) |
| $= \frac{1}{4}n^2(n+1)^2 - 2n$ | A1 | Correct expression |
| $= \frac{n}{4}(n^3 + 2n^2 + n - 8)$ | A1 | Completion to printed answer with no errors |

### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum_{r=20}^{50}(r^3-2) = \frac{50}{4} \times 130042 - \frac{19}{4} \times 7592$ | M1 | Attempt $S_{50} - S_{20}$ or $S_{50} - S_{19}$ and substitutes into correct expression at least once |
| $(= 1625525 - 36062)$ | A1 | Correct numerical expression (unsimplified) |
| $= 1\,589\,463$ | A1 | cao |

## Question 6(a):

**To show equivalence between $\frac{1}{4}k^2(k+1)^2 + (k+1)^3$ and $\frac{1}{4}(k+1)^2(k+2)^2$**

| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{4}k^2(k+1)^2 + (k+1)^3 = \frac{1}{4}k^4 + \frac{3}{2}k^3 + \frac{13}{4}k^2 + 3k + 1$ | M1 | Attempt to expand one correct expression up to a quartic |
| $\frac{1}{4}(k+1)^2(k+2)^2 = \frac{1}{4}k^4 + \frac{3}{2}k^3 + \frac{13}{4}k^2 + 3k + 1$ | M1 | Attempt to expand both correct expressions up to a quartic |
| One expansion completely correct | A1 | Dependent on both M1s |
| Both expansions correct and conclusion | A1 | |

**Or alternative method:**

| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{1}{4}(k+1)^2(k+2)^2 - \frac{1}{4}k^2(k+1)^2$ | M1 | Attempt to subtract |
| $= k^3 + 3k^2 + 3k + 1$ | M1 | Obtains a cubic expression |
| Correct expression | A1 | |
| $= (k+1)^3$, correct completion and comment | A1 | |

---