Question

Find the sum 1+3+5+7+9......+23.​

Answer 1

Let  =1+3+5+7+...+23

Flip the terms. It’s still the same sum.

=23+21+19+...+1

Add them together and you’ll get:

2∗=24+24+24+...+24

Notice what I’m doing here? Now those 24’s appear a certain amount of time which happens to be exactly the same amount as the number of terms in the sum. If you don’t know how many terms the sum has (like if you have just too many terms to count), you can conveniently use the properties of an arithmetic series where you can calculate any term as long as you know the first term and the constant you keep adding. In this case, the first term is 1, the last term is 23 and the constant is 2. Therefore:

23=1+2

23−1=2

=22/2=11

Don’t forget the first term! The actual amount is always plus one to that answer. Now we know how many terms the sum has, but it’s not over yet. In the beginning we added the sum together with itself, which is why the 2 appeared before   . Divide by 2 and we finally have:

=(12∗24)/2=144.

Answer 2

Answer:

144

Step-by-step explanation:

Sum of n odd nos = n^2

There are

$\frac{23 + 1}{2} = 12$

odd numbers in this case.

sum of first 12 odd numbers = 12^2 = 144.

Hope this helps you and you learnt something new....

If any doubts please comment....