What number is equal to the difference of the two sums?

Question

The sum of the first 25 multiples of 4 is: 4+8+12+...+100.

The sum of the first 25 multiples of 3 is: 3+6+9+...+75.

What number is equal to the difference of the two sums?

 

Answer 1

 

4+8+12+16+...+100

3+6+9+12+...+75

4-3=1

8-6=2

12-9=3

16-12=4

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100-75=25

you want the sum of the first 25 natural numbers

the sum of the first n natural numbers is n(n+1)/2=25(26)/2=25*13=325

also this is an arithmetic sequence...

S=(n/2)(2a+[n-1]d) where n=25, a=1(first term) and d=1 (common difference)

S=(25/2)(2+24)

S=(25/2)(26)=25*13=325 again

also the two original sets of multiples are also sequences so you could have used the formula for these as well

finally, a simple arithmetic solution looking at patterns...

1+25=26

2+24=26

3+23=26

and so on for a total of 12 groups of 26 plus the middle number of 13 gives...

12*26=312+13=325

 

Answer 2

Hi Young,

 

Used Excel to calculate the two sums (the first cell =4 and subsequent cells were an increase of the previous cell by 4)

 

The sum of the 25 multiples of 4 = 1300

 

Used Excel to calculate the two sums (the first cell =3 and subsequent cells were an increase of the previous cell by 3)

 

The sum of the 25 multiples of 3 = 975

 

The difference between 1300 and 975 = 325

 

Hope this helps...All the best,

Dr. D.