Question

Write the next line of this pattern b) Write the sequence of last numbers in each line. c) Write the first and last numbers of the 10 th line d) What will be the last number in the 9 th line?

Answer 1

Step-by-step explanation:

10 11 12 13

b) 1 4 9 13

c) and d) you can do it on your own

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Answer 2

Answer:

The next line of the series is 10     11     12    13     14     15     16.

b) The sequence of the last numbers is  1   4   9  16  25  36 and so on.

c) The first number of the $10^{th}$ line is 82.

    The last number of the $10^{th}$ line is 100.

d) The last number of the $9^{th}$ line is 81.

Step-by-step explanation:

In this series, the number of digits in line is following an increasing odd numbers pattern starting from 1,3,5, and so on.

• The first line has 1 number.

• The second line has 3 numbers.

• The third line has 5 numbers.

• The fourth line will have 7 numbers.

a) Since the numbers are consecutive numbers, the next numbers in the fourth line will be 7 numbers starting from 10

10     11     12    13     14     15     16

• Similarly, the first and last digits also follow the pattern in which each first number is incremented by an odd number starting from 1, 3, 5, and so on from their preceding line's first number.
• The last number is incremented by an odd number starting from 3, 5, 7, and so on from their preceding line's last number.

b) Sequence of last numbers in each line.

The following equation can be written for the last numbers.

x = previous row's last number + (2(n - 1) + 1)  

where n is the row number starting from 2.

The sequence of the last numbers is   1   4   9  16  25   36 and so on.

c) The first and last numbers of the $10^{th}$ line.

The following equation can be written for the first numbers.

x = previous row's first number + (2(n - 1) - 1)

where n is the row number starting from 2.

The first number of the $10^{th}$ line = the first number of the $9^{th}$ line + (2*9-1)

     = (2*9-1)+ (2*8-1) + (2*7-1) +(2*6-1)+(2*5-1)+(2*4-1)+(2*3-1)+(2*2-1)+(2*1-1) +1

The first number of the $10^{th}$ line = Sum of the first 9 odd numbers + 1

We know that the sum of first n odd natural numbers is $n^{2}$.

The sum of the first 9 odd numbers = 9*9 = 81.

The first number of the $10^{th}$ line is 82.

The last number of the $10^{th}$ line = the last number of the $9^{th}$ line + (2*9+1)

 = (2*9+1)+(2*8+1)+(2*7+1)+(2*6+1)+(2*5+1)+(2*4+1)+(2*3+1)+(2*2+1)+(2*1+1)+1

 = Sum of first 10 odd numbers

 = 10*10 = 100.

The last number of the $10^{th}$ line is 100.

d) The last number in the $9^{th}$ line.

Last number of $9^{th}$ line = Sum of first 9 odd numbers

                                      = 9*9 = 81

The last number of the $9^{th}$ line is 81.

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