Question #04343

2 Answers
Dec 9, 2017

333

Explanation:

First of all, we have that the sum of all the odd integers from 28 to 46 are 29 + 31 + ... + 43 + 45. That is a total of 9 numbers.

Let's represent this sum as a variable s:
s = 29 + 31 + ... + 43 + 45

Then reverse the order:
s = 45 + 43 + ... + 31 + 29

Add these two each other:
s + s = (29 + 45) + (31 + 43) + ... + (43 + 31) + (45 + 29)

s + s becomes 2s. Notice, from the left to right of the first two terms. As 29 increases by 2 to become 31, we can see that 45 also decreases by 2 to become 43. Since this goes on all the way, this means that essentisally, all of these terms are the same:
29 + 45 = 31 + 43 = ... = 43 + 31 = 45 + 29

And since there are 9 of them, we could just sum one of these terms and multiply by 9:
2s = (29 + 45) * 9 = 74 * 9 = 666.

Finally, we divide both sides by 2:
{2s}/2 = 666/2

s = 333

Dec 9, 2017

333

Explanation:

using Arithmatic progression
a_1 =29
a_n =45
we know that
a_n = a_1 + (n-1)d
45=29+(n-1)d
45-29=(n-1)d
16=(n-1)d {here d=2 because we have to count only odd number }
16/2=n-1
8=n-1
8+1=n

now, S_n=n/2 (a_1 +a_n)
S_9=9/2 (29+45)
S_9=9/2 (74)
S_9=9×37 = 333
hope ti helps