Question

For each number from $1\;{\text{to}}\;35$, how do you find all the ways to write it as a sum of two or more consecutive numbers?

Answer 1

Hint: First of all, limit the numbers to integers, all the numbers should necessarily have one way that is by adding consecutive positive and their respective negative numbers so that they get cancelled to zero and in final we get the number. Find numbers that are sum of consecutive numbers starting from one, in this method just add zero to them which will not affect the sum as well as give us one more way. Finally find numbers which are the sum of consecutive numbers not starting from $0,\;1$ and negative numbers.

Complete step-by-step solution:
To find all the ways to express the numbers from $1\;{\text{to}}\;35$ as sum of two or more consecutive numbers,
We will first get the sum with help of negative numbers, take an example can we express $4$ as following?
$4 = ( - 3) + ( - 2) + ( - 1) + 0 + 1 + 2 + 3 + 4$
In this way we can express all the numbers.
Now finding numbers that can be expressed as sum of consecutive numbers starting with $0$
$
  1 = 0 + 1 \\
  3 = 0 + 1 + 2 \\
  6 = 0 + 1 + 2 + 3 \\
  10 = 0 + 1 + 2 + 3 + 4 \\
  15 = 0 + 1 + 2 + 3 + 4 + 5 \\
  21 = 0 + 1 + 2 + 3 + 4 + 5 + 6 \\
  28 = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 \\
 $
Next number is $36$ that doesn’t belongs to domain
Except the first one we can write all of them, present in above series as
$
  3 = 1 + 2 \\
  6 = 1 + 2 + 3 \\
  10 = 1 + 2 + 3 + 4 \\
  15 = 1 + 2 + 3 + 4 + 5 \\
  21 = 1 + 2 + 3 + 4 + 5 + 6 \\
  28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 \\
 $
Now, sum starting from only positive numbers (not from $0\;{\text{and}}\;1$)
$
  5 = 2 + 3 \\
  9 = 2 + 3 + 4 \\
  14 = 2 + 3 + 4 + 5 \\
  20 = 2 + 3 + 4 + 5 + 6 \\
  27 = 2 + 3 + 4 + 5 + 6 + 7 \\
  35 = 2 + 3 + 4 + 5 + 6 + 7 + 8 \\
    \\
  7 = 3 + 4 \\
  12 = 3 + 4 + 5 \\
  18 = 3 + 4 + 5 + 6 \\
  25 = 3 + 4 + 5 + 6 + 7 \\
  33 = 3 + 4 + 5 + 6 + 7 + 8 \\
    \\
  9 = 4 + 5 \\
  15 = 4 + 5 + 6 \\
  22 = 4 + 5 + 6 + 7 \\
  30 = 4 + 5 + 6 + 7 + 8 \\
    \\
  11 = 5 + 6 \\
  18 = 5 + 6 + 7 \\
  26 = 5 + 6 + 7 + 8 \\
  35 = 5 + 6 + 7 + 8 + 9 \\
    \\
  13 = 6 + 7 \\
  21 = 6 + 7 + 8 \\
  30 = 6 + 7 + 8 + 9 \\
    \\
  15 = 7 + 8 \\
  24 = 7 + 8 + 9 \\
  34 = 7 + 8 + 9 + 10 \\
    \\
  17 = 8 + 9 \\
  27 = 8 + 9 + 10 \\
    \\
  19 = 9 + 10 \\
  30 = 9 + 10 + 11 \\
 $
$
  21 = 10 + 11 \\
  33 = 10 + 11 + 12 \\
    \\
  23 = 11 + 12 \\
    \\
  25 = 12 + 13 \\
    \\
  27 = 13 + 14 \\
    \\
  29 = 14 + 15 \\
    \\
  31 = 15 + 16 \\
    \\
  33 = 16 + 17 \\
    \\
  35 = 17 + 18 \\
 $
Therefore if we include negative numbers then we have following number of ways for each number respectively

Number	Number of ways
$1$	$1$
$2$	$1$
$3$	$3$
$4$	$1$
$5$	$3$
$6$	$3$
$7$	$3$
$8$	$1$
$9$	$5$
$10$	$3$
$11$	$3$
$12$	$1$
$13$	$3$
$14$	$3$
$15$	$7$
$16$	$1$
$17$	$3$
$18$	$5$
$19$	$3$
$20$	$3$
$21$	$5$
$22$	$3$
$23$	$3$
$24$	$3$
$25$	$5$
$26$	$1$
$27$	$7$
$28$	$3$
$29$	$3$
$30$	$7$
$31$	$3$
$32$	$1$
$33$	$7$
$34$	$1$
$35$	$7$

Note: You may have wondering that let us say for a number $15$, we have find only five ways then how does we written number of way to express it equals seven, see the following ways which are not visible to you in the solution:
$
  15 = ( - 6) + ( - 5) + ( - 4) + ( - 3) + ( - 2) + ( - 1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 \\
  {\text{And,}}\;15 = ( - 3) + ( - 2) + ( - 1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 \\
 $