| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Standard +0.3 This question requires applying standard formulas for sum to infinity of a GP and sum of n terms of an AP, then solving a linear equation. Part (b) is straightforward substitution into the nth term formula. While it involves two progressions, the required techniques are routine and the algebra is simple, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{5a}{1-(\pm\frac{1}{4})}\) | B1 | Use of correct formula for sum to infinity. |
| \(\frac{8}{2}[2a + 7(-4)]\) | \*M1 | Use of correct formula for sum of 8 terms and form equation; allow 1 error. |
| \(4a = 8a - 112\) leading to \(a = [28]\) | DM1 | Solve equation to reach a value of \(a\). |
| \(a = 28\) | A1 | Correct value. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(their\ 28 + (k-1)(-4) = 0\) | M1 | Use of correct method with *their* \(a\). |
| \([k =]\ 8\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{5a}{1-(\pm\frac{1}{4})}$ | B1 | Use of correct formula for sum to infinity. |
| $\frac{8}{2}[2a + 7(-4)]$ | \*M1 | Use of correct formula for sum of 8 terms and form equation; allow 1 error. |
| $4a = 8a - 112$ leading to $a = [28]$ | DM1 | Solve equation to reach a value of $a$. |
| $a = 28$ | A1 | Correct value. |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $their\ 28 + (k-1)(-4) = 0$ | M1 | Use of correct method with *their* $a$. |
| $[k =]\ 8$ | A1 | |
---4 The first term of an arithmetic progression is $a$ and the common difference is - 4 . The first term of a geometric progression is $5 a$ and the common ratio is $- \frac { 1 } { 4 }$. The sum to infinity of the geometric progression is equal to the sum of the first eight terms of the arithmetic progression.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.\\
The $k$ th term of the arithmetic progression is zero.
\item Find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q4 [6]}}