- Liz's method shows us the fact that the sum of three consecutive terms adds up
to a multiple of 3. By adding three consecutive numbers (3,4 and 5)visually by
using the respective numbers of squares. By moving a square from 5 down to
the column of three squares, she created a rectangle with side lengths 3 and 4,
proving the fact that the sum of any three consecutive numbers equals a
multiple of 3.
- Charlie's method of proving that the sum of any three consecutive terms equals
a multiple of 3 works by incrementing the smallest sequence of consecutive
numbers. With the procedure being repeated, again and again, the sums
increase by 3 (they become the next multiple of 3). By increasing the
sequence, again and again, proves that the pattern works for all three
consecutive terms.
When we add five consecutive numbers together, the sum is a multiple of 5. This can be proven algebraically as the five consecutive numbers can be represented algebraically as n, n+1, n+2, n+3, and n+4. Adding these terms together, and we get the formula to find the sum of the five consecutive numbers (5n+10, where n is the smallest number). Since the solution can be factorized by 5 to result in 5(n+2), this proves that the sun of five consecutive numbers equals a multiple of 5.
The same can be said to the sum of 7 consecutive terms(which when simplified algebraically equals 7n+21). The simplification can then be factorized as 7(n+3). Since the method of finding the sum of the seven consecutive terms involves multiplication by 7, the sum of seven consecutive terms equals a multiple of 7. This property applies to all the other odd number of consecutive terms as well.
However, the sum of an even number of consecutive terms (n) will always equal an odd number unless the even number of consecutive terms is a power of 2 greater than 2^1, as the factorization involves multiplying an even number to find the sum. Only the powers of 2 greater than 2^1, when such a number of consecutive terms are added together, will produce a result divisible by 2.