Consider the following sums of numbers and how they
are formed:
1 odd number
}
2 odd numbers
678
3 odd numbers
64748
Predict the following sum and complete the
generalization:
a. 1 + 3 + 5 + 7 + 9 + 11 = ?
b. 1 + 3 + 5 + Á + (2n - 1) = ?
Your examples do not fit your descriptions. For example,
678 is not "two odd numbers"
What are you trying to say?
90
To find the sum for question a, we need to add the given odd numbers together.
1 + 3 + 5 + 7 + 9 + 11 = ?
To find the sum for question b, we need to identify the pattern in the series and use the general formula for the sum of an arithmetic series.
In this series, we can observe that each number is odd and there is a consecutive pattern starting from 1. The next number is obtained by adding 2 to the previous number. The general formula for the nth term in this series is (2n - 1).
So, the sum of the series 1 + 3 + 5 + Á + (2n - 1) can be found using the formula for the sum of an arithmetic series. The sum of an arithmetic series is given by the formula:
S = (n/2) * (first term + last term)
In this case, the first term is 1 and the last term is (2n - 1).
Therefore, the sum of the series can be expressed as:
S = (n/2) * (1 + (2n - 1))
Now, we can use this formula to find the sum for any given value of n.