OCR MEI C2 — Question 1 12 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicArithmetic Sequences and Series
TypeTwo related arithmetic progressions
DifficultyStandard +0.3 This is a standard C2 sequences question testing routine application of arithmetic and geometric progression formulas. Part (i) involves solving simultaneous equations for A and D, then using the sum formula—straightforward algebra. Part (ii) requires manipulating the GP sum formula and solving a quadratic in r², both standard techniques with no novel insight required. Slightly above average difficulty due to the algebraic manipulation in part (ii), but still well within typical textbook exercises.
Spec1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
Question 1:
AnswerMarks Guidance
1(i) (A)
4A + 6D = 250 oe
D = 50,
AnswerMarks
A =  12.5 oeB1
B1
B1
B1
AnswerMarks Guidance
[4]condone lower-case a and d
1(i) (B)
2theirA49theirD [= 60 625] or
2
20
2their A19theirD [= 9250]
2
their “S  S ”
50 20
AnswerMarks
51 375 caoM1
M1
A1
AnswerMarks
[3]or a = their A + 20D
30
S  (al) oe with l = their A + 49D
AnswerMarks
30 230
S  (2their987.529their50)
30 2
AnswerMarks Guidance
1(ii) a  r2 1  a  r4 1 
=25 or 250
r1 r1
(r4 1)
a
r1 250
 oe
(r2 1) 25
a
(r1)
and completion to given result www
use of r4 1 = (r2 1)( r2 + 1) to obtain
r2 +1 = 10 www
r = ± 3
AnswerMarks
a = 6.25 or 12.5 oeB1
M1
M1
A1
A1
AnswerMarks
[5]at least one correct interim step required
or multiplication and rearrangement of
quadratic to obtain r4  10r2 + 9 = 0 oe with
all three terms on one side
or A1 for one correct pair of values of r and
AnswerMarks
aallow a(1 + r) as the denominator in the
quadruple- decker fraction
r2 = x oe may be used
or M1 for valid alternative algebraic
approaches eg using a(1 + r) = 25 and
ar2 + ar3 = ar2 (1 + r) = 225
or B2 for all four values correct, B1 for
both r values or both a values or one
pair of correct values if second M mark
not earned
Question 1:
1 | (i) | (A) | 2A + D = 25 oe
4A + 6D = 250 oe
D = 50,
A =  12.5 oe | B1
B1
B1
B1
[4] | condone lower-case a and d
1 | (i) | (B) | 50
2theirA49theirD [= 60 625] or
2
20
2their A19theirD [= 9250]
2
their “S  S ”
50 20
51 375 cao | M1
M1
A1
[3] | or a = their A + 20D
30
S  (al) oe with l = their A + 49D
30 2 | 30
S  (2their987.529their50)
30 2
1 | (ii) | a  r2 1  a  r4 1 
=25 or 250
r1 r1
(r4 1)
a
r1 250
 oe
(r2 1) 25
a
(r1)
and completion to given result www
use of r4 1 = (r2 1)( r2 + 1) to obtain
r2 +1 = 10 www
r = ± 3
a = 6.25 or 12.5 oe | B1
M1
M1
A1
A1
[5] | at least one correct interim step required
or multiplication and rearrangement of
quadratic to obtain r4  10r2 + 9 = 0 oe with
all three terms on one side
or A1 for one correct pair of values of r and
a | allow a(1 + r) as the denominator in the
quadruple- decker fraction
r2 = x oe may be used
or M1 for valid alternative algebraic
approaches eg using a(1 + r) = 25 and
ar2 + ar3 = ar2 (1 + r) = 225
or B2 for all four values correct, B1 for
both r values or both a values or one
pair of correct values if second M mark
not earned
\begin{enumerate}[label=(\roman*)]
\item An arithmetic progression has first term $A$ and common difference $D$. The sum of its first two terms is 25 and the sum of its first four terms is 250.

\begin{enumerate}[label=(\Alph*)]
\item Find the values of $A$ and $D$. [4]

\item Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
\end{enumerate}

\item A geometric progression has first term $a$ and common ratio $r$, with $r \neq \pm 1$. The sum of its first two terms is 25 and the sum of its first four terms is 250.

Use the formula for the sum of a geometric progression to show that $\frac{r^4 - 1}{r^2 - 1} = 10$ and hence or otherwise find algebraically the possible values of $r$ and the corresponding values of $a$. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C2  Q1 [12]}}
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