SOLUTION: what is the sum of the first 1,000 consecutive odd numbers?
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Question 259364: what is the sum of the first 1,000 consecutive odd numbers?
Found 4 solutions by Fombitz, stanbon, Alan3354, CharlesG2:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
The first 1000 consecutive odd numbers run from 1 to 1999.
The first 1000 consecutive even numbers run from 2 to 2000.
The sum of numbers from 1 to 2000 is,
Each even number is 1 greater than each odd number like
1, 2
3, 4
Each even number is an odd number plus one.
So then the total sum is the sum of all the odds twice plus 1000.
where OS is odds summed from 1 to 1999 and ES is the evens summed from 2 to 2000.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
what is the sum of the first 1,000 consecutive odd numbers?
---
1 + 3 + 5 + (2n-1) +.....+ (2*1000-1)
---
The 1000th term is 1999
----
Formula: S(n) = (n/2)(a(1) + a(n))
-------------------
S(1000) = (1000/2)(1+1999)
= 500(2000)
= 1,000,000
==================
Cheers,
Stan H.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
What is the relevance of consecutive?
How does that differ from the first 1000 odd numbers?
---------------------
The first 1000 consecutive odd numbers run from 1 to 1999.
Add the extremes:
1 + 1999 = 2000
3 + 1997 = 2000
All the pairs add to 2000
There are 500 pairs.
500*2000 = 1,000,000
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
what is the sum of the first 1,000 consecutive odd numbers?
consecutive numbers both even and odd table
(the sigma sign you see means to sum from n=0 to n=n)
n n(n+1)/2
1 1
2 3
3 6
4 10
5 15
10 55
100 5050
1000 500500
2000 2001000
now what is an odd number? an odd number can not be divisible by 2, we need to come up with a formula that only comes up with an odd number
2n-1 is an odd number where n=1,2,3,4,...
n=1 ---> 2n-1=2-1=1
n=2 ---> 2n-1=4-1=3
n=3 ---> 2n-1=6-1=5
n=4 ---> 2n-1=8-1=7
and so on to...
n=1000 --->2n-1=2000-1=1999
we want to sum the first 1000 consecutive odd numbers though
1+3=4=2^2
4+5=9=3^2
9+7=16=4^2
16+9=25=5^2
see what is happening here?
n 2n-1
1 1
2 3
3 5
4 7
5 9
10 19
100 199
1000 1999
n^2 plug in 1000 for n
1000^2=1000000 or 1 million
the sum of the first 1,000 consecutive odd numbers is 1 million
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