Question

How do you tell whether the sequence 1,3,5,7,15,31…….is geometric?

Answer 1

Hint: The above question is based on the geometric series. The main approach towards solving this kind of question is to find out whether every next term is obtained by multiplying or dividing the preceding number in the series. If the term which is multiplied or divided is the same throughout then the sequence is geometric.

Complete step-by-step solution:
A geometric sequence goes from one term to the next by always dividing or multiplying by the same number. The number multiplied or divided at each stage of a geometric sequence is called the "common ratio" r, because if you divide (that is, if you find the ratio of) successive terms, you'll always get the common value.
The above given sequence is \[1,3,5,7,15,31...\]
So, now we need to check whether it is geometric or not. So, first we need to find the common ratio between each term. For that general formula can be written as:
If the sequence is \[{a_1},{a_2},{a_3},{a_4}....\]
\[\dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{{a_3}}}{{{a_2}}} = 1\] which we call it as a ratio.
Now applying in the given sequence, we get to know that,
\[\dfrac{3}{1} \ne \dfrac{5}{3}\]
Here since the common ratio between the two consecutive terms are not the same. But multiplying every term with 2 and then adding 1 to it gives the next term. For example- \[1 \times 2 + 1 = 3\]i.e., 1 multiplied by 2 and added with 1 gives next term 3 which results in a geometric sequence.

Note: An important thing to note is that a geometric sequence will always have the same common ratio throughout i.e. either by multiplying or dividing the term to get the next term. We cannot only use addition or subtraction to find a common ratio.