A LevelFurther Maths: Core PureEdexcelTopic Questions9. Proof9.1 Proof by Induction

Proof by Induction (Edexcel A Level Further Maths: Core Pure)

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1
Sme Calculator6 marks

Prove by induction that for all positive integers n,

space f left parenthesis n right parenthesis equals 2 to the power of 3 n plus 1 end exponent plus 3 left parenthesis 5 to the power of 2 n plus 1 right parenthesis end exponent

is divisible by 17

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2
Sme Calculator6 marks

Prove by induction that for all positive integers n comma

open parentheses table row 3 0 row 6 1 end table close parentheses to the power of n equals open parentheses table row cell 3 to the power of n end cell 0 row cell 3 left parenthesis 3 to the power of n minus 1 right parenthesis end cell 1 end table close parentheses

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3
Sme Calculator6 marks

Prove by induction that for n element of straight integer numbers to the power of plus

sum from r equals 1 to n of left parenthesis 3 r plus 1 right parenthesis left parenthesis r plus 2 right parenthesis equals n left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis

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4
Sme Calculator6 marks

Prove by induction that for all positive odd integers n

space f left parenthesis n right parenthesis equals 4 to the power of n plus 5 to the power of n plus 6 to the power of n

is divisible by 15

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5
Sme Calculator6 marks

Prove by induction that for all positive integers n

space f left parenthesis n right parenthesis equals 3 to the power of 2 n plus 4 end exponent minus 2 to the power of 2 n end exponent

is divisible by 5

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