Yahoo Answers is shutting down on 4 May 2021 (Eastern Time) and the Yahoo Answers website is now in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Let [x] denote the greatest integer less than or equal to x. Find the value of [2/5]+[4/5]+[6/5]+...+[48/5]?
6 Answers
- ubiquitous_phiLv 71 decade agoFavourite answer
Greetings,
We must group the values by multiples of 5
2,4 < 5 so 2*0
6, 8, < 10 so 2*1
10, 12, 14 < 15 so 3*2
16, 18 < 20 so 2*3
20, 22, 24< 25 so 3*4
26, 28 < 30 so 2*5
30, 32, 34 < 35 so 3*6
36, 38 < 40 so 2*7
40, 42, 44 < 45 so 3*8
46, 48 < 50 so 2*9
2(1 + 3 + 5 + 7 + 9) + 3(2 + 4 + 6 + 8)
= 2*25 + 3*20
= 50 + 60
= 110
Regards
- 1 decade ago
If you're interested in just the answer, a program can quickly calculate this for you. If you're interested in how to obtain this answer, I suggest you review arithmethetic series.
--- Program Begin ---
0 + 0 + 1 + 1 + 2 + 2 + 2 + 3 + 3 + 4 + 4 + 4 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 8 + 8 + 8 + 9 + 9
sum = 110
--- Program End ---
Source(s): my programming skills: public class NumberTest { public static void main(String[] args) { int sum = 0; System.out.println("--- Program Begin ---"); for (int i=2; i<=48; i = i+2) { sum += f(i/5); System.out.print( f(i/5) + " + " ); } System.out.println("\n\nsum = " + sum + "\n\n" ); System.out.println("--- Program End ---"); } public static int f(double x) { int k = (int)Math.floor(x); return k; } } - Paschal HLv 61 decade ago
Notice that:
2/5 + 48/5 = 50/5 = 10
4/5 + 46/5 = 50/5 = 10
Get the idea?
I think there are 12 pairs like this, so 12*10 = 120
- cattbarfLv 71 decade ago
You can pair these up with greatest and least which are 2/5 + 48/5 = 10, and then the next pair
4/5 + 46/5 = 10, and keep going.
- vcs7578Lv 51 decade ago
apply the same concept that is applied to find sum of consecutive N natural numbers.
note [4/5]=0, [48/5]=9 etc. etc.
assume the sum is S. write it forward(as presented) and in reverse order.
S=[ 2/5]+[ 4/5]+[ 6/5]..............+[44/5]+[46/5]+[48/5]
S=[48/5]+[46/5]+[44/5]...........+[ 6/5] +[ 4/5] + [2/]
summing the two series
2S=(0+9)=(0+9)+(1+8) +(8+1)+(9+0)+(9+0)
There are 24 items of summation in the series.
2S=24*9
S=12*9=108
- ?Lv 45 years ago
often, [-x] + [x] = -a million if x isn't an integer and 0 if x is an integer. WLOG, we can assume that x is helpful. If x isn't an integer, then there exists an integer n such that n < x < n + a million wherein case, [x] = n. We be conscious that -n - a million < -x < -n. wherein case [-x] = -n - a million. hence [x] + [-x] = -a million whilst x is an integer, then [x] + [-x] = x + -x = 0.
