Standard +0.3 This is a straightforward arithmetic progression problem requiring students to find the common difference by equating consecutive term differences (2a - a = a² - 2a), solve the resulting quadratic (a² - 3a = 0), then apply the standard sum formula. While it involves multiple steps, each is routine and the problem type is standard for AS-level, making it slightly easier than average.
2 The first, second and third terms of an arithmetic progression are \(a , 2 a\) and \(a ^ { 2 }\) respectively, where \(a\) is a positive constant.
Find the sum of the first 50 terms of the progression.
May be done using 50th term \((=150)\). Their \(a\) and \(d\) must be numerical.
\(3825\)
A1
ISW. SC B2 for \(1275a\) or \(1275d\)
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $2a - a = a^2 - 2a$ | B1 | OE. An unsimplified correct equation in $a$ or $d$ only, e.g. $a^2 + a = 4a$. Can be implied by correct values for $a$ or $d$. |
| $a = 3$ or $d = 3$ | B1 | Condone 'extra' solution of $a = 0$ or $d = 0$. |
| $a = 3$ and $d = 3$ | B1 | SOI |
| $S_{50} = \frac{50}{2}(2 \times \text{their } a + 49 \times \text{their } d)$ | M1 | May be done using 50th term $(=150)$. Their $a$ and $d$ must be numerical. |
| $3825$ | A1 | ISW. **SC B2** for $1275a$ or $1275d$ |
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2 The first, second and third terms of an arithmetic progression are $a , 2 a$ and $a ^ { 2 }$ respectively, where $a$ is a positive constant.
Find the sum of the first 50 terms of the progression.\\
\hfill \mbox{\textit{CAIE P1 2022 Q2 [5]}}