Date | May 2022 | Marks available | 1 | Reference code | 22M.1.AHL.TZ1.10 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 1 |
Command term | Show that and Hence or otherwise | Question number | 10 | Adapted from | N/A |
Question
Consider the series ln x+p ln x+13ln x+…, where x∈ℝ, x>1 and p∈ℝ, p≠0.
Consider the case where the series is geometric.
Now consider the case where the series is arithmetic with common difference d.
Show that p=±1√3.
Hence or otherwise, show that the series is convergent.
Given that p>0 and S∞=3+√3, find the value of x.
Show that p=23.
Write down d in the form k ln x, where k∈ℚ.
The sum of the first n terms of the series is ln(1x3).
Find the value of n.
Markscheme
EITHER
attempt to use a ratio from consecutive terms M1
p ln xln x=13ln xp ln x OR 13ln x=(ln x)r2 OR p ln x=ln x(13p)
Note: Candidates may use ln x1+ln xp+ln x13+… and consider the powers of x in geometric sequence
Award M1 for p1=13p.
OR
r=p and r2=13 M1
THEN
p2=13 OR r=±1√3 A1
p=±1√3 AG
Note: Award M0A0 for r2=13 or p2=13 with no other working seen.
[2 marks]
EITHER
since, |p|=1√3 and 1√3<1 R1
OR
since, |p|=1√3 and -1<p<1 R1
THEN
⇒ the geometric series converges. AG
Note: Accept r instead of p.
Award R0 if both values of p not considered.
[1 mark]
ln x1-1√3 (=3+√3) (A1)
ln x=3-3√3+√3-√3√3 OR ln x=3-√3+√3-1 (⇒ln x=2) A1
x=e2 A1
[3 marks]
METHOD 1
attempt to find a difference from consecutive terms or from u2 M1
correct equation A1
p ln x-ln x=13ln x-p ln x OR 13ln x=ln x+2(p ln x-ln x)
Note: Candidates may use ln x1+ln xp+ln x13+… and consider the powers of x in arithmetic sequence.
Award M1A1 for p-1=13-p
2p ln x=43ln x (⇒2p=43) A1
p=23 AG
METHOD 2
attempt to use arithmetic mean u2=u1+u32 M1
p ln x=ln x+13ln x2 A1
2p ln x=43ln x (⇒2p=43) A1
p=23 AG
METHOD 3
attempt to find difference using u3 M1
13ln x=ln x+2d (⇒d=-13ln x)
u2=ln x+12(13ln x-ln x) OR p ln x-ln x=-13ln x A1
p ln x=23ln x A1
p=23 AG
[3 marks]
d=-13ln x A1
[1 mark]
METHOD 1
Sn=n2⌊2 ln x+(n-1)×(-13ln x)⌋
attempt to substitute into Sn and equate to ln(1x3) (M1)
n2⌊2 ln x+(n-1)×(-13ln x)⌋=ln(1x3)
ln(1x3)=-ln x3(=ln x-3) (A1)
=-3 ln x (A1)
correct working with Sn (seen anywhere) (A1)
n2⌊2 ln x-n3ln x+13ln x⌋ OR n ln x-n(n-1)6ln x OR n2(ln x+(4-n3)ln x)
correct equation without ln x A1
n2(73-n3)=-3 OR n-n(n-1)6=-3 or equivalent
Note: Award as above if the series 1+p+13+… is considered leading to n2(73-n3)=-3.
attempt to form a quadratic =0 (M1)
n2-7n-18=0
attempt to solve their quadratic (M1)
(n-9)(n+2)=0
n=9 A1
METHOD 2
ln(1x3)=-ln x3(=ln x-3) (A1)
=-3 ln x (A1)
listing the first 7 terms of the sequence (A1)
ln x+23ln x+13ln x+0-13ln x-23ln x-ln x+…
recognizing first 7 terms sum to 0 M1
8th term is -43ln x (A1)
9th term is -53ln x (A1)
sum of 8th and 9th term =-3 ln x (A1)
n=9 A1
[8 marks]
Examiners report
Part (a)(i) was well done with few candidates incorrectly using the value of p to verify rather than to 'show' the given result. In part (a)(ii) most did not consider both values of r and some did know the condition for convergence of a geometric series. Part (a)(iii) was generally well done but some had difficulty in simplifying the surd. Part (b) (i) and (ii) was generally well done. Although many completely correct answers to part b (iii) were noted, weaker candidates often made errors in properties of logarithms or algebraic manipulation leading to an incorrect quadratic equation.