Allow me to prove the result: odd numbers come in the form 2n-1, because 2n is always even, and the number immediately before an even number is always odd.

So, if we sum the first N odd numbers, we have

$\displaystyle \sum_{i=1}^N 2i-1 = 2\sum_{i=1}^N i - \sum_{i=1}^N 1$

The first sum is the sum of all integers from 1 to N, which is N(N+1)/2. We want twice this sum, so we have

$\displaystyle 2\sum_{i=1}^N i = 2\cdot\dfrac{N(N+1)}{2}=N(N+1)$

The second sum is simply the sum of N ones:

$\underbrace{1+1+1\ldots+1}_{N\text{ times}}=N$

So, the final result is

$\displaystyle \sum_{i=1}^N 2i-1 = 2\sum_{i=1}^N i - \sum_{i=1}^N 1 = N(N+1)-N = N^2+N-N = N^2$

which ends the proof.

5 0

2 years ago

8a2 = 12 + a plzzzzzzzzzzzzzzzzzzzzzzzzzz help me slove this problem

polet [3.4K]

Answer:

1.28

Step-by-step explanation:

Assuming in 8a2 the '2' represents an exponent, you answer should be

1.28

3 0

2 years ago

Select the correct answer. Why is it important to budget some money for entertainment? A. It is easier to stick to a budget if y

IRISSAK [1]

Answer:

Step-by-step explanation:

well some people like to go on a budget and others prefer not to because they rather spend their money money on wants rather than needs so I really hope this helps.

5 0

2 years ago