Help with Algebra homework

Question

Find the sum. Show all work and describe how to do it in words.

 

1. 1+2+3+...+171+172

 

2. 3+6+9+...+513+516

Answer 1

As David W. from Martinsburg put,

use Gauss' formula:

 

N(N + 1)/2

where N is the final number

 

3+6+9+...+513+516 = 

 

3(1+2+3+...+171+172) =

 

3[N(N+1)/2]

 

3[172(173)/2] = 44,634

 

 

Answer 2

Gauss' famous formula for the sum of the numbers from 1 to N is:    N(N+1)/2

 

For example:

    1   +   2   +   3    +    4   =   Sum

    4   +   3   +   2    +    1   =   Sum

--------------------------------------------

    5   +   5   +   5    +    5   =  2*Sum

 

    There are four 5's in twice the Sum.  So, Sum = 4*5/2 = 10

 

Here is another example:

   3   +   6   +   9   +  12   =  Total  =  3 * ( 1   +   2   +   3   +   4 )   =  3*Sum

  12  +   9   +   6   +   3    =  Total  =  3 * ( 4   +   3   +   2   +   1 )   =  3*Sum

--------------------------------------------------------------------------------------

  15  +  15  +  15  +  15   =   Total  = 3 * ( 5   +   5   +   5   +   5 )   =   3*(2*Sum)

 

   There are four 15's, or 3 times (4*5), in twice the Sum (that is, Total).  So, this makes 3*(4*5/2) for this Total.

 

After you answer the question, try this example:

What is this total?    51 + 52 + 53 + ... + 99 + 100   ?

Answer 3

Miss, here's the trick:

 

1 + 172 = 173

2 + 171 = 173

3 + 170 = 173

Etc...

 

There are half as many pairs, so there are 172/2 = 86 pairs.

 

173(86) = 14,878

 

I bet you can do the next one, but it's a little trickier!