Sum of Odd Numbers

Suppose the odd numbers are grouped in the following way: \{1\},\{3,5\},\{7,9,11\},\{13, 15, 17, 19\},\cdots What is the sum of the numbers in the tenth grouping?
Source: NCTM Mathematics Teacher 2006

SOLUTION
Brute force method
It doesn’t take long to write out the 60 consecutive odd numbers 1,3,5,7,\cdots,119 and notice that the tenth group consists of \{91,93,95,97,99,101,103,105,107,109\}. The sum of the numbers in the tenth grouping equals 1000.

Pattern method
Group 2 has 2 elements \{3,5\}
Value of first element =2(2-1)+1=3
Value of last element =3+(2-1)2=5
Sum of elements =3+5=8=2^3

Group 3 has 3 elements \{7,9,11\}
Value of first element =3(3-1)+1=7
Value of last element =7+(3-1)2=11
Sum of elements =7+9+11=27=3^3

Group 4 has 4 elements \{13,15,17,19\}
Value of first element =4(4-1)+1=13
Value of last element =13+(4-1)2=19
Sum of elements =13+15+17+19=64=4^3

In general group i has i elements
Value of first element =i(i-1)+1
Value of last element =i(i-1)+1+(i-1)2=(i-1)(i+2)+1
We prove that the sum of elements =i^3.
Recall that given n consecutive odd numbers \{a_1,a_2,a_3,\cdots,a_n\}
a_1+a_2+a_3+\cdots+a_n=\dfrac{(a_1+a_n)}{2}\times n
Sum of elements of group i=\dfrac{i(i-1)+1+(i-1)(i+2)+1}{2}\times i
=\dfrac{i^2-i+1+i^2+i-1}{2}\times i
=\dfrac{2i^2}{2}\times i
=i^3
Sum of the numbers in the tenth grouping
10^3=1000

Answer: 1000

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , . Bookmark the permalink.

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