6, 8, 10, 12, up tp 246. how long is the total length of all the rods?

Question

A support rod is 6 feet long and every rod after is 2 feet longer than the previous (6, 8, 10, 12.......246) up to 246. What is the formula to find the total length of all the rod lengths added together.

Answer 1

Apply Gauss' formula n(n+1)/2

6 to 246, but only the even numbers. 246/2 means 123 even numbers 2 to 246. Subtract 2, as it doesn't include the 2 or 4, to get 121 numbers in the sequence.

6,8,10,12,14......242,244,246

divide them all by 2 to get another sequence 3,4,5,....121,122,123

This is a sequence that can be summed applying Gauss' formula n(n+1)/2

If you sum 1 through 6, here n=6, the sum is 6(7)/2=21 =1+6 +2+5 +3+4 you can group then in subgroups adding to 7

Apply Gauss' formula to 1 through 123, then subtract off 1 and 2 which aren't in the sequence

sum=123(124)/2 = 123(62) =7626 subtract 1 and 2 to get the almost final answer: 7623. Now multiply it by 2 as we wanted the sequence double what we just found: 7623x2= 15,246 feet

Gauss came up with this formula when he was about 3 years old. His teacher wanted to keep him busy for a while and told him to add up all numbers from 1 to 100. In seconds Gauss gave the correct answer, with that formula n(n+1)/2 with n=100, so the sum was 100(101)/2=50(101)=5050. it took the teacher a while to grade him on that though. And the teacher was more careful with Gauss after that. There are 50 subgroups adding up to 101, 1+100, 2+99, ...50+51

Answer 2

 

Here's another way to look at the problem if you didn't know the formulas.

If the sequence started at 2, 2,4,6,8,...246, you would have 246/2=123 terms.

The sequence started at 6, so we have 123-2=121 terms.

The sequence is 6,8,10,12,...,240,242,244,246.

There are 121 terms. It is easy to find the 61st term: add and subtract the same number from 12 and 240 respectively until you find the middle term.

12+114=126 and 240-114=126, so 126 is the middle term.

Add the first and the last terms, add the second and the next to the last terms, and so on.

6+246=252, 8+244=252; you will always get 252.

There are 60 pairs that add up to 252(120/2=60 plus the middle term of 126)

60*252=15,120, 15,120+126=15,246 feet for the sum.

I would like to add something to my solution that came to me an hour ago. The average of all the numbers is 126; the average of all the pairs(252) is 126; the middle number therefore has to be 126-no work involved in determining the middle number !

Also, there are 60 groups of 252 or 120 groups of 126 plus the middle number which is 126 so we have 121 groups of 126:121*126=15,246 !

Answer 3

The formula for the nth term of an arithmetic sequence:

x_n = a + d(n-1)

 

xn is the nth term, a is the first term(6), d is the difference between terms(2)

 

246 is the nth term.

 

246 = 6 + 2(n-1)

246 = 6 + 2n - 2

246 = 4 + 2n

242 = 2n

n = 121

 

 

The formula for the sum of n terms:

S_n = (n/2) × (2a + (n-1)d)

 

(121/2)(2*6 + (121-1)2)

 

(121/2)(12 + 240)

 

(121/2)(252)

 

(121)(126) = 15,246

 

 

Answer 4

6+2(0)

6+2(1)

6+2+2=6+2(2)

6+2+2+2=6+2(3)

...

6+2(i)

...

6+2n = 246

2n = 240

n = 120

 

All rod lengths added together,

 

L = sum{i=0,120}(6+2i) =

 

sum{i=0,120}(6) + 2 sum{i=0,120}(i) =

 

121(6) + 2 (121)(120)/2 =

 

121(6 + 120) = 121*126 = 15,246 feet