Find a shortcut for computing the sum: 1+3+5+7++11+13+15+17=?? Explain your shortcut and how you could use this same shortcut to find the sum of all the odd numbers form 45 to 99.
what is the sum of the 1st and last?
what is the sum of 2nd and 2nd last ?
what is the sum of 3rd and 3rd last ?
mmmmhhhh?
1+17=18
3+15=18
5+13=18
7+11=18
4*18=72
45+99=144
47+97=144
49+95=144
51+93=144
53+91=144
55+89=144
57+87=144
59+85=144
61+83=144
63+81=144
65+79=144
67+77=144
69+75=144
71+73=144
14*144=2016
To find a shortcut for computing the sum of a series of odd numbers, we can make use of an arithmetic sequence formula. The formula for finding the sum of an arithmetic sequence is:
Sum = (n/2) * (first term + last term),
where n represents the total number of terms in the sequence.
In the given series, 1, 3, 5, 7, 9, 11, 13, 15, 17, we can observe that the first term is 1 and the common difference between consecutive terms is 2. The pattern is an arithmetic progression with a common difference of 2.
To find the number of terms, we can use the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference.
Solving for n, we can rearrange the formula as:
n = (nth term - first term + common difference) / common difference.
In this case, the nth term is 17, the first term is 1, and the common difference is 2, so substituting these values into our formula, we have:
n = (17 - 1 + 2) / 2,
n = (18) / 2,
n = 9.
Now we have the total number of terms, which is 9. Using the arithmetic sequence formula, we can calculate the sum:
Sum = (n/2) * (first term + last term),
Sum = (9/2) * (1 + 17),
Sum = 4.5 * 18,
Sum = 81.
Hence, the sum of the odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17 is 81.
To apply the same shortcut and find the sum of all the odd numbers from 45 to 99, we can follow these steps:
1. Calculate the number of terms (n) using the formula:
n = (last term - first term + common difference) / common difference.
In this case, the first term is 45, the last term is 99, and the common difference is 2.
n = (99 - 45 + 2) / 2,
n = 56.
2. Calculate the sum using the arithmetic sequence formula:
Sum = (n/2) * (first term + last term).
Sum = (56/2) * (45 + 99),
Sum = 28 * 144,
Sum = 4032.
Hence, the sum of all odd numbers from 45 to 99 is 4032.