Question

.How much will be the average of the squares of natural numbers from 1 to 40?​

Answer 1

Answer:

Thus, 20.5 is an average of first 40 natural numbers or positive integers.

Answer 2

Answer:

How much will be the average of the squares of natural numbers from 1 to 40?​

Explanation:

• What is average? A number expressing the central value from the given data set is known as average.
• What are Squares? When any given number is multiplied by itself and the result obtained is called as the square of the particular number.
• What are natural numbers?  All the positive numbers or integers sometimes zero is called as natural number.
• So the first natural numbers are 1,2, 3,...….,40.
• The square of first 40 natural number is-

1-1                                   21- 441

2-4                                 22- 484

3-9                                 23-529

4-16                                24-576

5-25                               25-625

6-36                               26-676

7-49                                27-729

8-64                                28-784

9-81                                 29-841

10-100                              30-900

11-121                                 31-961

12-144                               32-1024

13-169                                33- 1089

14-196                                34-1156

15-225                                35-1225

16-256                                36-1296

17-289                                 37-1369

18-324                                 38-1444

19-361                                   39-1521

20-400                                  40-1600

But using this method to solve is very hard and time consuming.

• There is the formula to solve this type of question.
• Sum of square of natural number is given by-$1^{2}+ 2^{2}+ 3^{2}+......+ 40^{2}$
• The formula is- $S_{40}=\frac{40(40+1)(2 \times 40+1)}{6}\\ = \frac{40\times 41(80+1)}{6} \\ = 20 \times 41 \times 27\\S_{40}= 22140$
• Average is given by= $\frac{S_{40} }{40}\\ \frac{22140}{40}\\ = 553.5$

Hence the average of the squares of natural numbers from 1 to 40 is 553.5

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