Standard +0.3 This is a standard geometric progression problem requiring manipulation of the sum formula and solving a quadratic equation. While it involves multiple steps (setting up S₄ and S₈, using the given ratio, solving for r, then finding S∞), the techniques are routine for A-level and the algebraic manipulation is straightforward. The 5-mark allocation confirms it's a moderate problem, slightly easier than average due to its predictable structure.
The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac{q}{p} = \frac{17}{16}\).
Find the two possible values of the sum to infinity. [5]
OE, i.e. substituting p and q expressions into ratio .
16
( 1−r4) ( 1−r8)
16=a , 17= a gets M0 unless recovered later.
(1−r) (1−r)
Answer
Marks
Guidance
Simplifying to 16r8−17r4+1 =0 (or equivalent form)
DM1
( 1−r8)
( 1+r4) 17
Or = = .
( 1−r4)
16
( 16r4 −1 )( r4 −1 ) =0 r= 1
Answer
Marks
Guidance
2
A1
1 1
Or r4 = r= (condone extra r=1solution).
16 2
10
S =
1
1−
Answer
Marks
Guidance
2
DM1
Use of correct sum to infinity formula with either of their r
values providing r 1.
20
S =20 and
Answer
Marks
Guidance
3
A1
Allow 6.67 or better.
A0 if there is only one or more than two S values.
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 6:
6 | ( 1−r8)
10
( 1−r8) ( 1−r4)
1−r 17 17
= [a = a ]
10 ( 1−r4) 16 (1−r) 16 (1−r)
1−r | M1* | 17
OE, i.e. substituting p and q expressions into ratio .
16
( 1−r4) ( 1−r8)
16=a , 17= a gets M0 unless recovered later.
(1−r) (1−r)
Simplifying to 16r8−17r4+1 =0 (or equivalent form) | DM1 | ( 1−r8)
( 1+r4) 17
Or = = .
( 1−r4)
16
( 16r4 −1 )( r4 −1 ) =0 r= 1
2 | A1 | 1 1
Or r4 = r= (condone extra r=1solution).
16 2
10
S =
1
1−
2 | DM1 | Use of correct sum to infinity formula with either of their r
values providing r 1.
20
S =20 and
3 | A1 | Allow 6.67 or better.
A0 if there is only one or more than two S values.
5
Question | Answer | Marks | Guidance
The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is $p$ and the sum of the first 8 terms of the progression is $q$. It is given that $\frac{q}{p} = \frac{17}{16}$.
Find the two possible values of the sum to infinity. [5]
\hfill \mbox{\textit{CAIE P1 2024 Q6 [5]}}