Sure, I'd be happy to help you break this down step by step:
### Part (iii)
The problem involves calculating the sum of a series of odd numbers:
[tex]\[
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 28
\][/tex]
Let's sum these numbers in sequence:
[tex]\[
1 + 3 = 4 \\
4 + 5 = 9 \\
9 + 7 = 16 \\
16 + 9 = 25 \\
25 + 11 = 36 \\
36 + 13 = 49 \\
49 + 15 = 64 \\
64 + 17 = 81 \\
81 + 19 = 100 \\
100 + 21 = 121 \\
121 + 28 = 149
\][/tex]
So, the total sum of these numbers is:
[tex]\[
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 28 = 149
\][/tex]
### Part (i)
We want to express 49 as the sum of 7 odd numbers. The first 7 odd numbers are:
[tex]\[
1, 3, 5, 7, 9, 11, 13
\][/tex]
Adding these together:
[tex]\[
1 + 3 + 5 + 7 + 9 + 11 + 13
\][/tex]
Calculating step by step:
[tex]\[
1 + 3 = 4 \\
4 + 5 = 9 \\
9 + 7 = 16 \\
16 + 9 = 25 \\
25 + 11 = 36 \\
36 + 13 = 49
\][/tex]
Therefore, 49 can be expressed as the sum of the first 7 odd numbers:
[tex]\[
49 = 1 + 3 + 5 + 7 + 9 + 11 + 13
\][/tex]
### Part (ii)
We want to express 121 as the sum of 11 odd numbers. The first 11 odd numbers are:
[tex]\[
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21
\][/tex]
Adding these together:
[tex]\[
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
\][/tex]
Calculating step by step:
[tex]\[
1 + 3 = 4 \\
4 + 5 = 9 \\
9 + 7 = 16 \\
16 + 9 = 25 \\
25 + 11 = 36 \\
36 + 13 = 49 \\
49 + 15 = 64 \\
64 + 17 = 81 \\
81 + 19 = 100 \\
100 + 21 = 121
\][/tex]
Therefore, 121 can be expressed as the sum of the first 11 odd numbers:
[tex]\[
121 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
\][/tex]
I hope this detailed solution helps clarify the problem! If you have any other questions, feel free to ask.
