Find the sum of all positive odd integers less than 100 by arithmetic progression
To find the sum of an arithmetic progression, we need to know the first term, the last term, and the common difference.
In this case, the first term is 1 (the smallest odd number), the common difference is 2 (since we are adding 2 to each term to get the next odd number), and the last term is 99 (the largest odd number less than 100).
We can find the number of terms (n) in this arithmetic progression using the formula:
n = (last term - first term) / common difference + 1
= (99 - 1) / 2 + 1
= 49
Now, we can find the sum of this arithmetic progression using the formula:
Sum = (n / 2)(first term + last term)
= (49 / 2)(1 + 99)
= 24.5 * 100
= 2450
Therefore, the sum of all positive odd integers less than 100 is 2450.
To find the sum of all positive odd integers less than 100 using an arithmetic progression, we need to determine the first term (a), the common difference (d), and the number of terms (n).
In this case, the first term (a) is 1 (since it is the first positive odd integer), the common difference (d) is 2 (since the difference between consecutive odd integers is always 2), and the last term is 99 (since it is the largest positive odd integer less than 100).
Using the formula for the sum of an arithmetic progression:
S = (n/2)(2a + (n-1)d)
We can calculate the sum (S) by substituting the values into the formula.
n = (99 - 1)/2 + 1
= 49 + 1
= 50
S = (50/2)(2*1 + (50-1)*2)
= 25(2 + 49*2)
= 25(2 + 98)
= 25(100)
= 2500
Therefore, the sum of all positive odd integers less than 100 is 2500.
To find the sum of all positive odd integers less than 100 using an arithmetic progression, we need to determine the first term, the common difference, and the number of terms in the progression.
In this case, the first term would be 1, as it is the smallest positive odd integer.
The common difference would be 2, as each subsequent term is obtained by adding 2 to the previous term to obtain the next odd integer.
To find the number of terms, we need to determine the highest odd integer less than 100. Since the odd integers less than 100 start at 1 and increase by 2, we can find the number of terms by dividing the difference between the highest and lowest odd integers by the common difference.
In this case, (99 - 1) / 2 = 49 is the number of terms.
Now, we can use the formula for the sum of an arithmetic progression:
Sum = (n / 2) * (2a + (n - 1) * d)
Where:
Sum is the sum of the arithmetic progression
n is the number of terms
a is the first term
d is the common difference
Using the values we found:
Sum = (49 / 2) * (2 * 1 + (49 - 1) * 2)
Simplifying the expression:
Sum = (49 / 2) * (2 + 48 * 2)
Sum = (49 / 2) * (2 + 96)
Sum = (49 / 2) * 98
Sum = 49 * 49
Sum = 2401
Therefore, the sum of all positive odd integers less than 100 using an arithmetic progression is 2401.