I really need help on these 2 math problems...?

Question

the problem is: 1, -1, 2, -2, 3, ___, ___

another one is: the sum of the first 100 positive even numbers

(u luk at da following table below to answer the question)

2 = 2 = 1 * 2

2 + 4 = 6 = 2 * 3

2 + 4 + 6 = 12 = 3 * 4

2 + 4 + 6 + 8 = 20 = 4 * 5

2 + 4 + 6 + 8 + 10 = 30 = 5 * 6

Answer 1

First is a simple pattern

1,-1,2,-2,3,-3,4

Perhaps you didnt figure out that the round peg goes into the round hole when you were young ?

The second one is just a factorial of 30 i believe....just google 30!.....its the same thing as doing 1x2x3x4...x30 from the pattern below...

Answer 2

-3 4  

Answer 3

-3,4

the second one is

2+4+6+8+10+12=42=6*7

(that is really interesting by the way)

Answer 4

1, -1, 2, -2, 3, -3, 4

2 = 2 = 1 * 2

2 + 4 = 6 = 2 * 3

2 + 4 + 6 = 12 = 3 * 4

2 + 4 + 6 + 8 = 20 = 4 * 5

2 + 4 + 6 + 8 + 10 = 30 = 5 * 6

2 + 4 + 6 + 8 + 10 + 12 = 42 = 6 * 7

2 + 4 + 6 + 8 + 10 + 12 + 14 = 56 = 7 * 8

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 = 72 = 8 * 9

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 90 = 9 * 10

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110 = 10 * 11

Answer 5

1, -1, 2, -2, 3, -3, 4

Notice its 1 then its negative, 2 then its negative, 3 then its negative, etc.

Notice its the (number of terms) * (number of terms  + 1)

so for 2 its 1 term, so 1*(1 + 1) = 2

for 2 + 4 its 2 terms, so 2* (2 + 1) = 6

So for the first 100 positive even numbers 100*(100 + 1) = 10100

Answer 6

-3, 4

10,100

Answer 7

Well, the first is a progression and continues with -3 and 4 for:

1, -1, 2, -2, 3, -3, 4

The second is a demonstration that a linear approach isn't always best. It is rumoured that Isaac Newton was once given a punishment at school where he had to add all the numbers from 1 to 100. He quickly realised that 1+100 = 2+99 = 3+98 = 4+97 and realised that there would be 50 such pairs totalling 101 and that 50 x 101 (5050) would be the answer.

The question you have posed, the sum of the first 100 positive even numbers, is strikingly similar:

2+200 = 4+198 = 6+196 and so on

50 pairs of numbers at 202 each yields: 10100