Date | May 2019 | Marks available | 5 | Reference code | 19M.2.SL.TZ2.S_10 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Find | Question number | S_10 | Adapted from | N/A |
Question
In an arithmetic sequence, u1=1.3 , u2=1.4 and uk=31.2.
Consider the terms, un, of this sequence such that n ≤ k.
Let F be the sum of the terms for which n is not a multiple of 3.
Find the exact value of Sk.
Show that F=3240.
An infinite geometric series is given as S∞=a+a√2+a2+…, a∈Z+.
Find the largest value of a such that S∞<F.
Markscheme
correct substitution (A1)
eg 3002(1.3+31.2) , 3002[2(1.3)+(300−1)(0.1)] , 3002[2.6+299(0.1)]
Sk=4875 A1 N2
[2 marks]
recognizing need to find the sequence of multiples of 3 (seen anywhere) (M1)
eg first term is u3 (= 1.5) (accept notation u1=1.5) ,
d=0.1×3 (= 0.3) , 100 terms (accept n=100), last term is 31.2
(accept notation u100=31.2) , u3+u6+u9+… (accept F=u3+u6+u9+…)
correct working for sum of sequence where n is a multiple of 3 A2
1002(1.5+31.2) , 50(2×1.5+99×0.3) , 1635
valid approach (seen anywhere) (M1)
eg Sk−(u3+u6+…) , Sk−1002(1.5+31.2) , Sk− (their sum for (u3+u6+…))
correct working (seen anywhere) A1
eg Sk−1635 , 4875 − 1635
F=3240 AG N0
[5 marks]
attempt to find r (M1)
eg dividing consecutive terms
correct value of r (seen anywhere, including in formula)
eg 1√2 , 0.707106… , a0.293…
correct working (accept equation) (A1)
eg a1−1√2<3240
correct working A1
METHOD 1 (analytical)
eg 3240×(1−1√2) , a<948.974 , 948.974
METHOD 2 (using table, must find both S∞ values)
eg when a=948 , S∞=3236.67… AND when a=949 , S∞=3240.08…
a=948 A1 N2
[5 marks]