Induction for series and sequences

Induction for series and sequences Quiz

Spec A1 · AQA A-Level Further Maths: Quiz

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5 questions, one answer each

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  1. 1. To go from SkS_k to Sk+1S_{k+1} in a sum of a series, you add

    • A. the kkth term
    • B. S1S_1
    • C. the (k+1)(k+1)th term
    • D. the formula with n=k+1n=k+1 substituted

  2. 2. Evaluate ∑r=15(2r−1)\sum_{r=1}^{5}(2r-1).

    • A. 2525
    • B. 99
    • C. 1010
    • D. 4545

  3. 3. Which is the correct way to begin the inductive step for ∑r=1nr=n(n+1)2\sum_{r=1}^{n}r=\frac{n(n+1)}{2}?

    • A. Assume the result for n=k+1n=k+1
    • B. Check n=2n=2 only
    • C. Substitute n=k+1n=k+1 into the left-hand side only
    • D. Assume the result for n=kn=k, then add (k+1)(k+1) to both sides

  4. 4. In the inductive step for ∑r=1nr=n(n+1)2\sum_{r=1}^{n}r=\frac{n(n+1)}{2}, k(k+1)2+(k+1)\frac{k(k+1)}{2}+(k+1) simplifies to

    • A. k(k+2)2\frac{k(k+2)}{2}
    • B. (k+1)(k+2)2\frac{(k+1)(k+2)}{2}
    • C. (k+1)22\frac{(k+1)^2}{2}
    • D. k(k+1)(k+2)2\frac{k(k+1)(k+2)}{2}

  5. 5. u1=1u_1=1 and un+1=2un+1u_{n+1}=2u_n+1. Which formula gives unu_n for all positive integers nn?

    • A. 2n−12n-1
    • B. n2n^2
    • C. 2n−12^n-1
    • D. 2n+12^n+1

Quiz