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Question: How do you tell whether the sequence \[100,50,25,\dfrac{25}{2},\dfrac{25}{4},.....\] is geometric?...

How do you tell whether the sequence 100,50,25,252,254,.....100,50,25,\dfrac{25}{2},\dfrac{25}{4},..... is geometric?

Explanation

Solution

In this problem, we have to find whether the given sequence is a geometric sequence. Here for geometric sequence we have a term called common ratio, which is the ratio of the successive term and the preceding term. We have to find the common ratio for the given sequence and if every common ratio is the same, then the given sequence will be a geometric sequence.

Complete step by step solution:
We know that the given sequence is,
100,50,25,252,254,.....100,50,25,\dfrac{25}{2},\dfrac{25}{4},.....
We can now assume variables for every term in the sequence,
Where,
a1=100,a2=50,a3=25,a4=252,a5=254{{a}_{1}}=100,{{a}_{2}}=50,{{a}_{3}}=25,{{a}_{4}}=\dfrac{25}{2},{{a}_{5}}=\dfrac{25}{4}
We have to find whether the given sequence is a geometric sequence, for which it has constant common ratios.
Now we have to find the common ratio for every term.
We know that the common ratio is the ratio of the successive term and the preceding term.
We can find the common ratio for first two terms, we get
⇒a2a1=50100=0.5\Rightarrow \dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{50}{100}=0.5
In such a way, we can find the common ratio, we get

& \Rightarrow \dfrac{{{a}_{3}}}{{{a}_{2}}}=\dfrac{25}{50}=0.5 \\\ & \Rightarrow \dfrac{{{a}_{4}}}{{{a}_{3}}}=\dfrac{\dfrac{25}{2}}{25}=\dfrac{25}{50}=0.5 \\\ & \Rightarrow \dfrac{{{a}_{5}}}{{{a}_{4}}}=\dfrac{\dfrac{25}{4}}{\dfrac{25}{2}}=\dfrac{2}{4}=0.5 \\\ \end{aligned}$$ We can now see that every common ratio is the same. $$\Rightarrow {{a}_{1}}={{a}_{2}}={{a}_{3}}={{a}_{4}}={{a}_{5}}=0.5$$ Here we have the same common ratios. **Therefore, the given sequence $$100,50,25,\dfrac{25}{2},\dfrac{25}{4},.....$$ is a geometric sequence.** **Note:** Students make mistakes while finding the common ratio, which is the ratio of every successive term to its respective preceding term. We should always remember that, only if the common ratios are equal for every term, the equation is said to be geometric. There we need to know the difference between geometric sequence and arithmetic sequence.