Let m be the side length of the fourth smallest possible square with integer value of its area, and let n be the sum of all prime numbers less than 10. Find MN(N-M).

510

Step-by-step explanation:

It's assumed 0 is not an integer.

M is obviously 2. The area of a square is the square of its side length. If this area must be an integer, then it must be the fourth integer, which is 4. The side length,

A prime number is a positive integer which can be divided exactly by only itself and 1. In fact, it has only those two factors. The last statement excludes 1 as a prime number. Hence, the prime numbers less than 10 are 2, 3, 5 and 7. Their sum is .

To evaluate ,

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}