# Expression for sum of numbers

#### qeb

1: 3,5: 7,9,11 ....sum of these groups can be expressed as n raised to 3. How can I prove this?

#### Plato

MHF Helper
1: 3,5: 7,9,11 ....sum of these groups can be expressed as n raised to 3. How can I prove this?
$n^3:~1,~8,~27,~64,~\cdots$ add numbers the given groups.

Is the next group $13,~15,~17,~19~?$

#### Debsta

MHF Helper
Interesting.
Group 1 : 1 Sum 1 : 1
Group 2 : 3,5 Sum 2 : 8
Group 3 : 7, 9,11 Sum 3 : 27
Group 4 : 13,15,17,19 Sum 4 : 64
etc
Each group is an arithmetic progression.
If the group number is x, then the first number in the group is x(x-1)+1
Use the sum of an arithmetic progression formula with a=x(x-1)+1, d=2 and n = x (seeing that the group number is the same as the number of terms in the group)
Then do some algebra. Falls out nicely!

(Not sure if this constitutes a "real" proof as a=x(x-1)+1 was found by observation.)

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1 person

#### Archie

Nicely done.
(Not sure if this constitutes a "real" proof as a=x(x-1)+1 was found by observation.)
Note that the first $$\displaystyle n$$ groups use $$\displaystyle \sum \limits_{k=1}^n k$$ odd numbers. So the first number of each group is the $$\displaystyle \left(1+\sum \limits_{k=1}^n k\right)$$th odd number. You may well already know the formula for the sum of the first $$\displaystyle n$$ natural numbers (1, 2, 3, ...). this will give your value for $$\displaystyle a$$.