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Maths-

General

Easy

Question

If A = $open square brackets table row 1 0 row 1 1 end table close square brackets$ or $open square brackets table row 1 0 row 0 1 end table close square brackets$ , then which of the following holds for all n ≥ 1, by principle of mathematical induction

Aⁿ = nA – (n– 1) I
Aⁿ = 2^n–1 A– (n–1) I
Aⁿ = nA + (n –1) I
Aⁿ = 2^n–1 A+ (n–1) I

The correct answer is: Aⁿ = nA – (n– 1) I

Let A = $open square brackets table row 1 0 row 1 1 end table close square brackets$
$rightwards double arrow$ Aⁿ = nA – (n –1)I
$rightwards double arrow$ A = nA – (n –1) A = A which is true
If A = $open square brackets table row 1 0 row 1 1 end table close square brackets$ $rightwards double arrow$ A² = $open square brackets table row 1 0 row 2 1 end table close square brackets$ = A
Using Mathematical Induction,
A^m+1 = A^m. A= (mA –(m–1)I) A
= mA² –mA + A
= m $open square brackets table row 1 0 row 2 1 end table close square brackets$ –mA + $open square brackets table row 1 0 row 1 1 end table close square brackets$
= m $open square brackets table row 1 0 row 2 1 end table close square brackets$ – m $open square brackets table row 1 0 row 1 1 end table close square brackets$ + $open square brackets table row 1 0 row 1 1 end table close square brackets$ + $open square brackets table row 1 0 row 0 1 end table close square brackets$ m – m $open square brackets table row 1 0 row 0 1 end table close square brackets$
= $open square brackets table row cell m minus m plus 1 plus m end cell 0 row cell 2 m minus m plus 1 plus 0 end cell cell m minus m plus 1 plus m end cell end table close square brackets$ – m $open square brackets table row 1 0 row 0 1 end table close square brackets$
= $open square brackets table row cell m plus 1 end cell 0 row cell m plus 1 end cell cell m plus 1 end cell end table close square brackets$ – m $open square brackets table row 1 0 row 0 1 end table close square brackets$
= (m +1)A –mA which is also true.
Thus choice (A) is true for both values of A.
If (A) is possible then (C) can’t be true. Again (B) and (d) are not possible (they have no symmetricity).
Choice (A) is correct.

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