Question

40, 10, 5/2 5/8... (fractions) is it arithmetic or geometric and what are the next two terms PLZ HELP DUE IN 10 MINS

Answer 1

Please check the explanation.

Explanation:

Given the sequence

`40,\:10,\:(5)/(2),\:(5)/(8)`

A geometric sequence has a constant ratio 'r' and is defined by

`\:a_n=a_0\cdot r^(n-1)`

Computing the ratios of all the adjacent terms

`(10)/(40)=(1)/(4),\:\quad ((5)/(2))/(10)=(1)/(4),\:\quad ((5)/(8))/((5)/(2))=(1)/(4)`

The ratio of all the adjacent terms is the same and equal to

`r=(1)/(4)`

Thus, the given sequence is a geometric sequence.

As the first element of the sequence is

`a_1=40`

Therefore, the nth term is calculated as

`\:a_n=a_0\cdot r^(n-1)`

`a_n=40\left((1)/(4)\right)^(n-1)`

Put n = 5 to find the next term

`a_5=40\left((1)/(4)\right)^(5-1)`

`a_5=40\cdot (1)/(4^4)`

`a_5=(40)/(4^4)`

`=(2^3\cdot \:5)/(2^8)`

`a_5=(5)/(2^5)`

`a_5=(5)/(32)`

now, Put n = 6 to find the 6th term

`a_6=40\left((1)/(4)\right)^(6-1)`

`a_6=40\cdot (1)/(4^5)`

`a_6=(40)/(4^5)`

`=(2^3\cdot \:5)/(2^(10))`

`a_6=(5)/(2^7)`

`a_6=(5)/(128)`

Thus, the next two terms of the sequence 40, 10, 5/2, 5/8... is:

• 
  `a_5=(5)/(32)`

• 
  `a_6=(5)/(128)`