what is the 114th term in 0.123456....

Question

I want to teach a student how to find the nth decimal number for the decimal 0.12345678.....

Answer 1

The pattern 123456790 is recurring. This is equivalent to the fraction 10/81. The series can be written as a sum=∑(r/10ʳ).

So the pattern contains 9 digits and the zero appears every 9th digit. The 108th digit is therefore zero. That makes the 114th digit 6.

Here is the reasoning:

10/81=10(1/9)²=10(1/(10-1))²=10[1/(10(1-0.1))²]=0.1(1-0.1)⁻².

The binomial expansion of this is 0.1(1+2/10+((-2)(-3)/2!)/100-((-2)(-3)(-4)/3!)/1000+...=

0.1(1.23456...).

This reduces to 0.1∑(r/10ʳ⁻¹)=∑(r/10ʳ) which is the given series.