Who's the smartest?

Question

Alona told her son Frank to paint the fence. Frank whined and complained. So she said " OK, you dont have to do it today. Instead paint 1/2 of one board today. Tomorrow paint 1/3 of a board and 2/3 of a board. I dont care which board you paint. The next day paint 1/4 of a board, 2/4 of a board, and 3/4 of a board. That will cover 3 days; continue like that for another 22 days. I'll do the rest." If you add up all the boards Frank completed and the parts he painted, Frank paint the eqivalent of how many boards?



Be sure to answer with steps/ exlpanation

Answer 1

Okay, on day 1, he paints 1/2 of a board, on day 2, he paints (1+2)/3 of a board, on day 3, he paints (1+2+3)/4 of a board, and on day n, we can assume that he paints [k=1, n]∑k/(n+1) of a board. So on all 25 days, he paints a grand total of:

[n=1, 25]∑[k=1, n]∑k/(n+1)

Let us resolve this sum. Using the Gaussian formula for the sum of the first n positive integers:

[n=1, 25]∑(n(n+1)/2)/(n+1)

Simplifying:

[n=1, 25]∑n/2
1/2 [n=1, 25]∑n

Using the Gaussian formula again:

1/2 (25)(26)/2
162.5 boards.

And we are done.

Answer 2

Day 1: ½ board
Day 2: 1/3 + 2/3 board = 3/3 = 1 board
Day 3: 1/4 + 2/4 + 3/4 board = 6/4 = 1½ boards
Day 4: 1/5 + 2/5 + 3/5 + 4/5 = 10/5 = 2 boards
Day 5: 1/6 + 2/6 + 3/6 + 4/6 + 5/6 = 15/6 = 2½ boards

Do you see a pattern. Each day he does another ½ board.

If you double the amount he does, the sequence would be:
1 + 2 + 3 + 4 + 5 + ... + 21 + 22 + 23 + 24 + 25

The formula for this sum is:
n(n+1) / 2
= 25 * 26 / 2
= 25 * 13
= 325

But he did half of this each day, so divide by 2:
After 25 days he would have painted 162½ boards

Answer 3

uhh....
1 1/2= .5
2 1/3 + 2/3= 1
3 1/4 + 2/4 + 3/4 = 1.5
4 1/5 + 2/5 + 3/5 + 4/5 = 2
follow the pattern........
so .5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 + 4.5 + 5 + 5.5 + 6 + 6.5 + 7 + 7.5 + 8 + 8.5 +9 + 9.5 + 10 + 10.5 + 11 = ?

Answer 4

first day ------1/2 ----------------1/2
second day----1/3+2/3 =>2/2
third day -------1/4+2/4+3/4 =>3/2
---
----
nth day --------1/(n+1)+2/(n+1) =>n/2

so total painting completed in n days

1/2+2/2+3/2+4/2.............n/2

=> 1/2[1 + 2 +3 .............n]

=>1/2[n(n+1)/2] (since sum of first n numbers = n(n+1)/2)

substitute n = 25

1/2[(25(26)/2 ] = 162.5 boards