squares and reciprocals?

Question

please show work if possible

1. how many squares of various sizes are in s six-by-six square?

2. find the sum of the reciprocals of the first five positive powers of three. express your answer as a common fraction

Answer 1

QUESTION 1:

Think of the individual squares of size 1... there are 6² = 36 of those.

Now think of individual squares of size 2... there are 5² = 25 of those.

Continue the pattern.
Size 3 --> 4² = 16 of them
Size 4 --> 3² = 9 of them
Size 5 --> 2² = 4 of them
Size 6 --> 1² = 1 (the whole square).

Add them up and you get:
1 + 4 + 9 + 16 + 25 + 36 = 91 squares total.

QUESTION 2:
This question is a little confusing. By the first five positive powers of 3, do they mean 3^1, 3^2, 3^3, 3^4 and 3^5? That would be my assumption.

(Others might interpret it as 1^3, 2^3, 3^3, etc. but the method would be the same. These aren't powers of 3 but instead the first five positive *cubes*)

The next problem is do they mean 1, 3, 9, 27, 81, or do they mean 3, 9, 27, 81, 243?

In the first case the results are positive. In the second case the exponents are positive. Again, I'll use the second case, but you should be able to do this with the other numbers, if that is what your question is looking for.

1, 3, 9, 27, 81

Reciprocals are 1/n:
1/1, 1/3, 1/9, 1/27, 1/81

The common denominator would be the last number (81):
1/1 = 81/81
1/3 = 27/81
1/9 = 9/81
1/27 = 3/81
1/81 = 1/81

Adding you get 121/81 = 1 40/81.

(If you did this with 3, 9, 27, 81, 243 the answer would be 121/243).

Answer 2

1. too many to count the squares... i think there's a program that can do it for you.. not sure though.

2. i don't understand the question. what do u mean by first positive powers of 3?

Answer 3

Oh Sir Joey, do no longer use large words with us Aussies.... it hurts our brains. noticeably with the rugby international cup coming, the great we are able to purely be able to take care of for the subsequent months or so is preserving beer and reckoning on the television demonstrate and yell and scream. different primes in different bases? PRIMES in different bases!!??? hard sufficient coping with base ten, mate! do you want us to advance extra hands and ft or something?