The sequence of odd numbers starting with 3 is 3,5,7,9...
Consecutive numbers from this sequence can be added using the following pattern-
3+5+7+9=4x6
3+5+7+9+11=5x7
3+5+7+9+11+13=6x8
The first n numbers in this sequence are added. Find a formula for the total.
2007-09-07 04:28:15 · 13 answers · asked by Unknown! 2 in Science & Mathematics ➔ Mathematics
3+5+7+9+...+2k+1= 1+2+3+...+2k+1- 1 -(2+4+...+2k)
= (2k+1)(2k+2)/2-1-2(1+2+...+k)
= (k+1)(2k+1)-k(k+1)-1
= (k+1)(k+1)-1
=(k+1)^2-1 = k(K+2)
2007-09-07 04:34:01 · answer #1 · answered by Anonymous · 1⤊ 1⤋
If you add the numbers the answer is the same as if you would multiply the result. In other words the result of:
3+5+7+9=
is the same as:
4x6
which is 24
The rest are the same.
3+5+7+9=4x6 (24)
3+5+7+9+11=5x7 (35)
3+5+7+9+11+13=6x8 (48)
2007-09-07 04:35:24 · answer #2 · answered by acydskull 4 · 1⤊ 0⤋
the number of numbers determines =4, =5,=6 etc. ie 3+5=2.......
then when numbers are added tog they make x (3+5+7). x=
the sum of 3+5+7........ x another number
ie 3+5+7+9= 4 because there are 4 numbers to be added
times ? which makes both sides equal.
sorry this is a rubbish explanation, I know what I mean but i'm not a mathmatician.
2007-09-07 04:37:50 · answer #3 · answered by cairn4lodge 4 · 0⤊ 0⤋
Start in another way.
1. the sum of the first odd number is 1
2. the sum of the 2 first odd numbers is 4
3. the sum of the 3 first odd numbers is 9
4.the sum of the 4 first odd numbers is 16
5.the sum of the 5 first odd numbers is 25
continue and you see that tne sum of the n first odd numbers is n x n.
Then you might see that the sum of the n first numbers is:
(n/2 x n/2) x2+ n i think.
example: the sum of the 20 first numbers:
10 x 10 x 2 + 10= 210
First number is 1.
Just try.
2007-09-07 08:07:51 · answer #4 · answered by anordtug 6 · 0⤊ 0⤋
You are already most of the way there. You have noticed that the sum of the numbers is a certain multiple (in each case) of the number of numbers involved; have you seen that this multiple is always 2 higher than the number of numbers?
This means the total is always 'n' multiplied by n+2; i.e. total = n^2 + 2n.
That's all there is to it, I hope it helps.
Please feel free to drop me a line if you'd like any further explanation.
2007-09-07 05:17:12 · answer #5 · answered by general_ego 3 · 0⤊ 0⤋
N X (N+2)
2007-09-07 04:32:29 · answer #6 · answered by Matthew O 5 · 1⤊ 2⤋
there's a clue interior the contemporary learn you have made at school. My wager is which you have been finding at a thank you to remedy communities of equations with various unknown portions. precise? The form of miles travelled is composed of two factors, that on the city and that on the line. that's your first equation. The time is in a similar fashion composed.[2d] Speeds on the city and on highway are given and those are distance/time respectively.[third and 4th] we are as much as six unknowns and easily 4 equations so a techniques, so we decide for 2 greater equations. i wish you will see what those are [hint: you should apply the three.7 hours (an person-friendly one) and the 8.5 gallons (particularly trickier)]
2016-10-10 03:23:11 · answer #7 · answered by Anonymous · 0⤊ 0⤋
Consider.
S = 1+3+5+..+nth term+(n+1)th term
S=(n+1)^2 obv... (sum of odd numbers 1 to n is just n^2)
So your series S'=S-1= (n+1)^2-1=n(n+1)
2007-09-07 12:46:33 · answer #8 · answered by alienfiend1 3 · 0⤊ 0⤋
I suggest that you throw odd number 1 into the summation too.
Then the answer will be a lot more elegant.
2007-09-07 04:35:34 · answer #9 · answered by Alexander 6 · 3⤊ 0⤋
For the first 'n' numbers the fromula is
n*(n+2).
2007-09-10 19:47:42 · answer #10 · answered by defeNder 3 · 0⤊ 0⤋