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Date May 2018 Marks available 6 Reference code 18M.1.SL.TZ2.S_7
Level Standard Level Paper Paper 1 Time zone Time zone 2
Command term Find Question number S_7 Adapted from N/A

Question

An arithmetic sequence has u1=logc(p)u1=logc(p) and u2=logc(pq)u2=logc(pq), where c>1c>1 and p,q>0p,q>0.

Let p=c2p=c2 and q=c3q=c3. Find the value of 20n=1un20n=1un.

Markscheme

METHOD 1 (finding u1u1 and d)

recognizing =S20=S20 (seen anywhere)      (A1)

attempt to find u1u1 or d using logcck=klogcck=k     (M1)
eg  logcclogcc3logcc3logcc, correct value of u1u1 or d

u1u1 = 2, d = 3 (seen anywhere)      (A1)(A1)

correct working     (A1)
eg  S20=202(2×2+19×3),S20=202(2+59),10(61)S20=202(2×2+19×3),S20=202(2+59),10(61)

20n=1un20n=1un = 610     A1 N2

 

METHOD 2 (expressing S in terms of c)

recognizing =S20=S20 (seen anywhere)      (A1)

correct expression for S in terms of c      (A1)
eg  10(2logcc2+19logcc3)10(2logcc2+19logcc3)

logcc2=2,logcc3=3logcc2=2,logcc3=3  (seen anywhere)     (A1)(A1)

correct working      (A1)

eg  S20=202(2×2+19×3),S20=202(2+59),10(61)S20=202(2×2+19×3),S20=202(2+59),10(61)

20n=1un20n=1un = 610     A1 N2

 

METHOD 3 (expressing S in terms of c)

recognizing =S20=S20 (seen anywhere)      (A1)

correct expression for S in terms of c      (A1)
eg  10(2logcc2+19logcc3)10(2logcc2+19logcc3)

correct application of log law     (A1)
eg  2logcc2=logcc4,19logcc3=logcc57,10(logc(c2)2+logc(c3)19),10(logcc4+logcc57),10(logcc61)2logcc2=logcc4,19logcc3=logcc57,10(logc(c2)2+logc(c3)19),10(logcc4+logcc57),10(logcc61)

correct application of definition of log      (A1)
eg  logcc61=61,logcc4=4,logcc57=57logcc61=61,logcc4=4,logcc57=57

correct working     (A1)
eg  S20=202(4+57),10(61)S20=202(4+57),10(61)

20n=1un20n=1un = 610     A1 N2

[6 marks]

 

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series
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