Question

How do you find the next two terms of the geometric sequence $405,\,135,\,45,\,...$ ?

Answer 1

First, we will learn about geometric sequence. A geometric sequence is a sequence or progression which contains non-zero numbers. In this sequence, each term is calculated by multiplying the previous number with a fixed common factor.

Formula used:
${a_{i + 1}} = {a_i} \times r$

Complete step by step answer:
According to the question, the sequence is a geometric sequence. So, we will try to solve the question by using the geometric sequence formula. The formula is:
${a_{i + 1}} = {a_i} \times r$
Here $r$ is the common factor. Now, we need to find $r$. For that, we need to keep $r$ alone. So, we will shift the term ${a_i}$to the other side of the equation, and we get:
$\Rightarrow r = \dfrac{{{a_{i + 1}}}}{{{a_i}}}$
Now, we will put the values of the question according to the formula, and we get:
$\Rightarrow r = \dfrac{{{a_{i + 1}}}}{{{a_i}}}$
$\Rightarrow r = \dfrac{{135}}{{405}} = \dfrac{{45}}{{135}}$
When we simplify the equation, we get:
$\Rightarrow r = \dfrac{1}{3}$
Therefore, we get that $r$ is $\dfrac{1}{3}$.
Now, we know that the starting term here is ${a_1}$. So, we get that:
${a_1} = 405$
The corresponding terms are ${a_2},\,{a_3},\,{a_4},\,{a_5},...$ and so on. To calculate the values of these terms, we need to apply the formula:
${a_{i + 1}} = {a_i} \times r$
When we calculate the values of these terms, we get:
${a_2} = 405 \times \dfrac{1}{3}$
$\Rightarrow{a_2} = 135$
Similarly, we can calculate ${a_3},\,{a_4},\,{a_5},\,...$, and we get:
${a_3} = 135 \times \dfrac{1}{3}$
$\Rightarrow{a_3} = 45$
$\Rightarrow{a_4} = 45 \times \dfrac{1}{3}$
$\Rightarrow{a_4} = 15$
$\Rightarrow{a_5} = 15 \times \dfrac{1}{3}$
$\Rightarrow{a_5} = 5$
So, now we can write the formula as:
${a_n} = 405 \times {\left( {\dfrac{1}{3}} \right)^{n - 1}}$

Therefore, the next two numbers of the geometric sequence $405,\,135,\,45,\,...$ is $15\,\,and\,\,5$.

Note: Many students make a mistake by calculating the wrong common factor that is $r$, and most students get confused when $r$ is a fractional part. This question was easy, but there are difficult problems also. So, we need to always check whether the number is consistently true by multiplying the common ratio by other terms. This helps us verify the answer.