Let T(k) be the statement 1+3+5+ +(2 k-1)=k^ 2 +10 Which of the following is correct?

Correct answer: B. T(k) is true T(k+1) is true When k=1, LHS=1 but RHS=1+10=11 T(1) is not true Let T(k) is true. That is array l 1+3+5+ .+(2 k-1)=k^ 2 +10 Now, 1+3+5+ ..+(2 k-1)+(2 k+1) =k^ 2 +10+2 k+1=(k+1)^ 2 +10 array T(k+1) is true. That is T(k) is true T(k+1) is true. But T(n) is not true for all n ∈ N, as T(1) is not true.

BITSAT 2012

Exam: BITSAT
Year: 2012
Subject: Mathematics
Chapter: Mathematical Induction

BITSAT 2012 · Mathematics · Mathematical Induction

Let T(k) be the statement 1+3+5+ +(2 k-1)=k^ 2 +10 Which of the...

Let

\mathrm{T}(k)

be the statement

1+3+5+\ldots+(2 k-1)=k^{2}+10

Which of the following is correct?

Options

A $\mathrm{T}(1)$ is true
B $\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true
C $\mathrm{T}(n)$ is true for all $n \in \mathbf{N}$
DAll above are correct

Correct answer

B. $\mathrm{T}(k)$ is true $\Rightarrow \mathrm{T}(k+1)$ is true

Step-by-step solution

When

\mathrm{k}=1, \mathrm{LHS}=1

but

\mathrm{RHS}=1+10=11\therefore \mathrm{T}(1)

is not true Let

\mathrm{T}(k)

is true. That is

$\begin{array}{l} 1+3+5+\ldots.+(2 k-1)=k^{2}+10 \\ \text{ Now, } 1+3+5+\ldots..+(2 k-1)+(2 k+1) \\ =k^{2}+10+2 k+1=(k+1)^{2}+10 \end{array}

\therefore \mathrm{T}(k+1)

is true. That is

\mathrm{T}(k)

is true

\Rightarrow \mathrm{T}(k+1)

is true. But

\mathrm{T}(n)

is not true for all

n \in \mathbf{N},

as

\mathrm{T}(1)$ is not true.

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