Challenging +1.2 This is a standard Further Maths summation problem requiring the method of differences or direct summation of a cubic polynomial. While it involves algebraic manipulation across multiple steps (expanding the general term, applying summation formulas for powers, and factorizing), the technique is well-practiced in Further Maths courses. The 6-mark allocation and straightforward structure place it above average difficulty but well within expected Further Maths competency.
Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series
$$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$
Express your answer as a product of linear factors. [6]
A cubic equation either has 3 real roots or 1 real root. Since the sum of squares is negative, the 3 roots cannot all be real so there is just 1 real root.
B1
AO2
Question 4(b)
Answer
Marks
Guidance
A second root is \(1 - 3i\), since complex roots occur in conjugate pairs.
B1
AO2
The third root is 2 since the sum of the 2 complex roots is 2 and the sum of the 3 roots is 4.
E1, B1
AO2, AO1
Total: [7]
$$S_n = \sum_{r=1}^{n} r(r+1)(r+3)$$ | M1 | AO1
$$= \sum_{r=1}^{n} (r^3 + 4r^2 + 3r)$$ | A1 | AO1
$$= \frac{n^2(n+1)^2}{4} + 4\frac{n(n+1)(2n+1)}{6} + 3\frac{n(n+1)}{2}$$ | A1 | AO1
$$= \frac{n(n+1)}{12}(3n(n+1) + 8(2n+1) + 18)$$ | A1 | AO1
$$= \frac{n(n+1)}{12}(3n^2 + 19n + 26)$$ | A1 | AO1
$$= \frac{n(n+1)(n+2)(3n+13)}{12}$$ | A1 | AO1
| **Total: [6]** |
# Question 4(a)
$$\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\beta\gamma + \gamma\alpha + \alpha\beta)$$ | M1 | AO2
$$= 4^2 - 2 \times 14 = -12$$ | A1 | AO2
A cubic equation either has 3 real roots or 1 real root. Since the sum of squares is negative, the 3 roots cannot all be real so there is just 1 real root. | B1 | AO2
# Question 4(b)
A second root is $1 - 3i$, since complex roots occur in conjugate pairs. | B1 | AO2
The third root is 2 since the sum of the 2 complex roots is 2 and the sum of the 3 roots is 4. | E1, B1 | AO2, AO1
| **Total: [7]** |
Find an expression, in terms of $n$, for the sum of the first $n$ terms of the series
$$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$
Express your answer as a product of linear factors. [6]
\hfill \mbox{\textit{WJEC Further Unit 1 Q3 [6]}}