Question

Find the average of squares of the first 11 consecutive even numbers.
A.184
B.148
C.186
D.174

Answer 1

Hint: In these types of questions first determine the first even numbers which can be determined since the all even numbers are the multiple of number 2 or exactly divisible by 2 after finding the even numbers you can directly use the squaring operation and find the average of the 11 even numbers.

Complete step-by-step answer:
Let’s find out the first consecutive 11 even numbers
Since we know that the number that are multiple of 2 or exactly divisible by 2 are called even numbers
So the 11 consecutive multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and 22
Let’s find the square of the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and 22

${2^2} = 2 \times 2$	4
${4^2} = 4 \times 4$	16
${6^2} = 6 \times 6$	36
${8^2} = 8 \times 8$	64
${10^2} = 10 \times 10$	100
${12^2} = 12 \times 12$	144
${14^2} = 14 \times 14$	196
${16^2} = 16 \times 16$	256
${18^2} = 18 \times 18$	324
${20^2} = 20 \times 20$	400
${22^2} = 22 \times 22$	484

So the average of the square of the 11 consecutive even numbers is equal to $Average = \dfrac{{{S_T}}}{{{T_N}}}$ here ${S_T}$ is the sum of the total numbers and ${N_T}$is the total numbers present
${S_T}$= 4 + 16 + 36 + 64 + 100 + 144 + 196 + 256 + 324 + 400 + 484
$ \Rightarrow $${S_T}$ = 2024
We know that the ${N_T}$ = 11 since we are finding the average of 11 consecutive numbers
Substituting the value of ${S_T}$ and ${N_T}$ in the average formula i.e. $Average = \dfrac{{{S_T}}}{{{T_N}}}$
$Average = \dfrac{{2024}}{{11}}$
$ \Rightarrow $Average = 184
Hence the average of the squares of first consecutive even numbers is equals to 184

Note: Here we discussed about the even number now let’s talk about the odd numbers and some properties shown by even and odd numbers so the numbers which are not multiple and are not exactly divisible by 2 are called odd numbers but odd numbers shows some special properties i.e. whenever we add two odd numbers the result comes an even number whereas the sum of 2 even number result an even number but the addition of an odd and even number is gives odd number in result.