Question

a) i) Give an inductive formula for the sum of the first $ n $ odd numbers: \[ 1+3+5+\ldots+2 n-1 \] Show your induction process. ii) Use mathematical induction to prove the correctness of your inductive formula in i) above. $ [(4)+(4)] $ b) The following $ \mathrm{C}++ $ program segment computes $ \mathbf{C}=\mathbf{A} * \mathbf{B} $, where $ \mathbf{A} $ is an " $ \mathrm{m} $ " by " $ \mathrm{n} $ " matrix, $ \mathbf{B} $ is an " $ \mathrm{n} $ " by " $ \mathrm{r} $ " matrix and $ \mathbf{C} $ is an " $ \mathrm{m} $ " by " $ \mathrm{r} $ " matrix. for (long int $ i=0 ; \mathrm{i}<\mathrm{m} ; \mathrm{i}++ $ ) for (long int $ j=0 ; \mathrm{j}<r ; j++ $ ) \{ long double Sum =0; for (long int $ k=0 ; k<n ; k++ $ ) Sum $ = $ Sum $ +A[i][k] $ * $ B[k][j] ; $ $ C[i][]]= $ Sum for $ (l o $ fo \{ \} \} \} How many times does the code iterate in computing i) Sum? ii) The $ j^{\text {th }} $ column of $ \mathbf{C} $, i.e. $ \mathbf{C}[\mathrm{i}][\mathrm{j}] $ for a fixed $ \mathrm{i} $ and $ \mathrm{j}=0,1,, 2, \ldots, \mathrm{r}-1 $ ? iii) The complete matrix $ \mathbf{C} $ ? $ [2+4+6] $

          a) i) Give an inductive formula for the sum of the first \( n \) odd numbers:
\[
1+3+5+\ldots+2 n-1
\]
Show your induction process.
ii) Use mathematical induction to prove the correctness of your inductive formula in i) above.
\( [(4)+(4)] \)
b) The following \( \mathrm{C}++ \) program segment computes \( \mathbf{C}=\mathbf{A} * \mathbf{B} \), where \( \mathbf{A} \) is an " \( \mathrm{m} \) " by " \( \mathrm{n} \) " matrix, \( \mathbf{B} \) is an " \( \mathrm{n} \) " by " \( \mathrm{r} \) " matrix and \( \mathbf{C} \) is an " \( \mathrm{m} \) " by " \( \mathrm{r} \) " matrix.
for (long int \( i=0 ; \mathrm{i}<\mathrm{m} ; \mathrm{i}++ \) )
for (long int \( j=0 ; \mathrm{j}<r ; j++ \) )
\{
long double Sum =0;
for (long int \( k=0 ; k<n ; k++ \) )
Sum \( = \) Sum \( +A[i][k] \) * \( B[k][j] ; \)
\( C[i][]]= \) Sum
for \( (l o \)
fo
\{
\}
\}
\}
How many times does the code iterate in computing
i) Sum?
ii) The \( j^{\text {th }} \) column of \( \mathbf{C} \), i.e. \( \mathbf{C}[\mathrm{i}][\mathrm{j}] \) for a fixed \( \mathrm{i} \) and \( \mathrm{j}=0,1,, 2, \ldots, \mathrm{r}-1 \) ?
iii) The complete matrix \( \mathbf{C} \) ?
\( [2+4+6] \)

$a) i) Give an inductive formula for the sum of the first n odd numbers: 1+3+5+…+2 n-1 Show your induction process. ii) Use mathematical induction to prove the correctness of your inductive formula in i) above. [(4)+(4)] b) The following C++ program segment computes 𝐂=𝐀 * 𝐁, where 𝐀 is an " m " by " n " matrix, 𝐁 is an " n " by " r " matrix and 𝐂 is an " m " by " r " matrix. for (long int i=0 ; i<m ; i++ ) for (long int j=0 ; j<r ; j++ ) { long double Sum =0; for (long int k=0 ; k<n ; k++ ) Sum = Sum +A[i][k] * B[k][j] ; C[i][]]= Sum for (l o fo { } } } How many times does the code iterate in computing i) Sum? ii) The j^th column of 𝐂, i.e. 𝐂[i][j] for a fixed i and j=0,1,, 2, …, r-1 ? iii) The complete matrix 𝐂 ? [2+4+6]$

Added by Rajuili B.

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