Q154 · Practice questionTechnology-freeComplex familiar7 marks
QUESTION 154 (7 marks)
A sequence of partial sums is defined by
Here denotes the factorial of the positive integer .
a)[1 mark]
Calculate exactly.
b)[4 marks]
Prove by mathematical induction that for every positive integer .
c)[2 marks]
Determine the least for which . Justify your answer.
WORKED SOLUTION
7 marksPractice marking scheme
ANSWER
(a) . (b) for all positive integers . (c) .
Worked solution
(a) .
(b) For , , so the statement holds.
Assume for an arbitrary positive integer that . Then
This is the required formula with . By mathematical induction the statement holds for every positive integer .
(c) requires . Since and , the least is 6. Factorials increase here, so all earlier values fail.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Obtain 23/24.
Part b: Verify the base case.
Part b: State the hypothesis for an arbitrary positive integer m.
Part b: Prove the m+1 formula by adding the next term.
Part b: Conclude for all positive integers by induction.
Part c: Convert the strict inequality to (n+1)!>1000.
Part c: Compare 6! and 7! and conclude n=6.
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
View the QCAA syllabusHow many marks did you earn?
Compare your working with the guide above.