Question

Is every power of 10 from 100 onwards, a term of the arithmetic sequence
19, 28, 37, ...?​

Answer 1

The answer of the following question is given below-

Arithmetic sequence given is 19, 28, 37,....?

First find the difference between each value.

consider values as a1, a2, a3 and so on.

a2-a1 i.e., 28-19=9

a3-a2 i.e. 37-28=9

Use formula {An=a1+(n-1)d}

An=19+(n-1)9

a1=19 (first number)

An=37 (last Nth number)

d=9 (difference of all consecutive members)

n=3 number of values.

Sum of finite series values-

Definition: The sum of the values of a finite arithmetic continuation is known an arithmetic series.

19+28+37

This sum can be obtained by proceeding the number N of terms being calculated via adding (here 3), then multiplying by the sum of the first and last number in the continuation (here 19 + 37 = 56), and dividing by 2:

n(a1+An)/2, i. e., 3(19+37)/2

The sum of the 3 values of this series is 84.

This series correlates to the succeeding straight line y=9x+19

Obtaining the Nth element-

a1 =a1+(n-1)*d =19+(1-1)*9 = 19

a2 =a1+(n-1)*d =19+(2-1)*9 = 28

a3 =a1+(n-1)*d =19+(3-1)*9 = 37

a4 =a1+(n-1)*d =19+(4-1)*9 = 46

a5 =a1+(n-1)*d =19+(5-1)*9 = 55

a6 =a1+(n-1)*d =19+(6-1)*9 = 64

a7 =a1+(n-1)*d =19+(7-1)*9 = 73

a8 =a1+(n-1)*d =19+(8-1)*9 = 82

a9 =a1+(n-1)*d =19+(9-1)*9 = 91

And so on.

Arithmetic sequence will be-

19, 28, 37, 46, 55, 64, 73, 82, 91......Nth element.