Question= Find the sum of all the odd numbers up to and including 99,999.
Would a=1 because its the first odd number in the sequence? And would d=2 ?
So then would you do S99,999= 49999.5(2+99,998(2))
Then divide by two?
yes, a=1,d=2 but there are 50000 terms, so
S = 50000/2 (2+49999(2))
or, since you know the first and last term,
S = 50000/2 (1+99999)
there are 50,000 odd numbers
The sum of the first n odd numbers is n^2.
Thank you !
To find the sum of all the odd numbers up to and including 99,999, we need to first identify the sequence of odd numbers.
In this case, the first odd number in the sequence is 1, and the common difference between consecutive odd numbers is 2. Therefore, yes, a = 1 and d = 2.
To find the sum, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Here,
- Sn is the sum of the first n terms of the sequence.
- n is the number of terms in the sequence.
- a is the first term.
- d is the common difference.
Applying this formula to our case, we have:
- n = (99,999 - 1) / 2 + 1 = 50,000 (since we are considering odd numbers only)
- a = 1
- d = 2
Substituting these values into the formula:
Sn = (50,000/2)(2*1 + (50,000-1)*2)
= 25,000(2 + 99,998*2)
= 25,000(2 + 199,996)
= 25,000(199,998)
Hence, the sum of all odd numbers up to and including 99,999 is:
Sn = 25,000(199,998) = 4,999,500,000.