Mathematics · May / June 2022 · Paper 21 · Question 6
YMA01
As printed in the paper
The question as text
6. In a geometric sequence u1 , u2 , u3 , … • the common ratio is r • u2 + u3 = 6 • u4 = 8
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
6(a)[3]
u
=
ar
n
−
1
ar
+
a
r
2
=
6
and
ar =
3
8
n
ar
+
ar
2
6
3
=
+
1
r
=
r
2
ar
3
8
4
3
r −
2
4
r
−
4
=
0
*
(a)
8
8
u
=
8
u
=
,
u
=
Way 2
4
3
r
2
r
2
8
8
+
=
6
r
r
2
3
r −
2
4
r
4
0
*
−
=
2
6(b)[3]
r = −
3
8
ar
3
=
8
a
=
=
...
3
2
−
3
u = −
27
1
−
27
6(c)[2]
S∞
=
=
...
2
1
−−
3
81
= −
5
Notes:
(a) Ignore labelling and mark (a), (b) and (c) together
M1: Uses the correct nth term formula for a GP to set up two equations in a and r
May also be in terms of u1 and r or u2 and r or u3 and r e.g.
u
u
r
+
u r
2
=
6,
u r
3
=
8
or
u
+
u r
=
6,
u r
2
=
8 or
3
+
u
=
6,
ru
=
8
1
1
1
2
2
2
r
3
3
Must be using the correct term formula so e.g.
ar
2
+
ar
3
=
6
and
ar =
4
8
is M0
1
−
r
3
Alternatively may use the correct sum formula for the first equation,
oe.
a
−
a
=
6
1
−
r
M1: Attempts to solve their two equations to get an equation in r. Look for an attempt to divide the
two equations, or an attempt to find a in terms of r from one and substitute into the other.
Allow slips but the algebra should essentially be correct so do not allow use of e.g. ar3 = (ar)3
2
2
a
4
a
Alternatively, attempts to eliminate r from the equation,
r
=
+
=
6
3
a
3
a
3
a
2
2
1
1
2
a
3
+
4
a
3
=
6
. Award when they reach a quadratic in
a in this case.
3
A1*: CSO Note that as we are marking (a), (b) and (c) together, allow the printed answer to appear
anywhere as long as it follows correct work but the “=0” must be seen not implied.
Way 2:
M1: Uses u4 = 8 to write u2 and u3 in terms of r
M1: Uses u2 + u3 = 6 to get an equation in r
A1*: CSO
(b)
B1: Correct value of r seen or used in their working even if subsequently rejected and ignore any
other value offered e.g. r = 2
M1: Uses a value for r from solving the equation given in (a) where |
r < in one of their equations
| 1
from (a) to find a value for u1 or a. Allow slips but the algebra should essentially be correct
8
A1ft: Correct value and no other values. Follow through on
for a value of r with |
r <
| 1
(
)
their
r
3
(c)
M1: Uses the correct sum formula with their a (u1) obtained from |
r < and r where |
| 1
r <, to find the sum
| 1
to infinity.
A1: Correct answer and no other values. Allow equivalents e.g. −16.2
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