Problem Solving

lkcr wrote:1. 2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8 =?

Why does that equal 2^9? Am I missing something.....
(^ = to the power of)

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8
= 2 + 2(1 + 2 + 2² + ... + 2^7)
Now, 1 + 2 + 2² + ... + 2^7 is a geometric series, where first term = 2 and the common ratio = 2 > 1

Geometric Series is of the form: a + ar + ar² + ... + ar^(n - 1)
Here, a = first term
r = common ratio
n = number of terms
Then sum of n terms of geometric series = a(r^n - 1)/(r - 1) when r > 1 and a(1 - r^n)/(1 - r) when r < 1

In the given case, 1 + 2 + 2² + ... + 2^7 = 1(2^8 - 1)/(2 - 1) = 2^8 - 1

Therefore, 2 + 2(1 + 2 + 2² + ... + 2^7) = 2 + 2(2^8 - 1) = 2 + 2^9 - 2 = 2^9 = 512

lkcr wrote: 2. (1001^2-999^2) /(101^2-99^2)

What would be the best approach to handle 2.?

Sorry for these silly questions..

We know that a² - b² = (a - b)(a + b), so apply this identity in the given question.
(1001² - 999²)/(101² - 99²) = (1001 - 999)(1001 + 999)/(101 - 99)(101 + 99)
= (2)(2000)/(2)(200)
= 2000/200
= 10