UniGeniusA-level Further Maths · Edexcel 9FM0

CP 1.1

Induction for sums and divisibility

The lesson

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1nr(r+2)=16n(n+1)(2n+7)\sum_{r=1}^{n} r(r + 2) = \frac{1}{6}n(n + 1)(2n + 7) for all positive integers nn.
[5]
(b)
Hence find ∑r=1120r(r+2)\sum_{r=11}^{20} r(r + 2).
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1n(2r−1)=n2\sum_{r=1}^{n} (2r - 1) = n^2 for all positive integers nn.
[5]
(b)
Hence find ∑r=1130(2r−1)\sum_{r=11}^{30} (2r - 1).
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1nr⋅r!=(n+1)!−1\sum_{r=1}^{n} r \cdot r! = (n + 1)! - 1 for all positive integers nn.
[5]
(b)
Hence find ∑r=16r⋅r!\sum_{r=1}^{6} r \cdot r!.
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1nr3=14n2(n+1)2\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2 for all positive integers nn.
[5]
(b)
Hence find ∑r=610r3\sum_{r=6}^{10} r^3.
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1n1(2r−1)(2r+1)=n2n+1\sum_{r=1}^{n} \frac{1}{(2r - 1)(2r + 1)} = \frac{n}{2n + 1} for all positive integers nn.
[5]
(b)
Hence find ∑r=1201(2r−1)(2r+1)\sum_{r=1}^{20} \frac{1}{(2r - 1)(2r + 1)}.
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1n(3r−1)=12n(3n+1)\sum_{r=1}^{n} (3r - 1) = \frac{1}{2}n(3n + 1) for all positive integers nn.
[5]
(b)
Hence find ∑r=2140(3r−1)\sum_{r=21}^{40} (3r - 1).
[2]

7 marksCore techniqueShow full working

(a)
Prove by induction that ∑r=1nr⋅2r−1=(n−1)2n+1\sum_{r=1}^{n} r \cdot 2^{r-1} = (n - 1)2^n + 1 for all positive integers nn.
[5]
(b)
Hence find ∑r=18r⋅2r−1\sum_{r=1}^{8} r \cdot 2^{r-1}.
[2]

5 marksCore techniqueShow full working

Prove by induction that f(n)=32n−1f(n) = 3^{2n} - 1 is divisible by 88 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=5n+3f(n) = 5^n + 3 is divisible by 44 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=23n−1f(n) = 2^{3n} - 1 is divisible by 77 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=11n−6f(n) = 11^n - 6 is divisible by 55 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=6n+4f(n) = 6^n + 4 is divisible by 55 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=4n+6n−1f(n) = 4^n + 6n - 1 is divisible by 99 for all positive integers nn.
[5]

5 marksCore techniqueShow full working

Prove by induction that f(n)=n3−nf(n) = n^3 - n is divisible by 66 for all positive integers nn.
[5]

Independent practice for Pearson Edexcel A-level Further Mathematics (9FM0), not endorsed by Pearson.

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