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Question: What are the next three terms of the sequence \(1,-3,9,-27...\) ?...

What are the next three terms of the sequence 1,−3,9,−27...1,-3,9,-27... ?

Explanation

Solution

The first thing that we have to do is to observe the sequence carefully. We can see that the sequence contains only the multiples of 33 . Clearly this is a geometric sequence. But we need to precisely find the common ratio. So, knowing the first and second terms of the sequence, we put them in the general expression. Knowing the common ratio, we put n=5,6,7n=5,6,7 one by one to get the next three terms.

Complete step by step answer:
A sequence is an enumerated collection of objects, especially numbers, in which repetitions are allowed and in which the order of objects matters. A sequence may be finite or infinite depending on the number of objects in the sequence. Sequences can be of various types such as arithmetic sequence, geometric sequence and so on. Sequences can be completely random as well. The nth{{n}^{th}} term of a sequence is sometimes written as a function of n.
A geometric sequence is the one in which the ratio between the consecutive objects is a constant. It is called the common ratio of the geometric sequence. The general expression of the nth{{n}^{th}} term will be,
an=arn−1....(1){{a}_{n}}=a{{r}^{n-1}}....\left( 1 \right) where, a is the first term of the sequence and r is the common ratio.
In this problem, the first term is 11 and the second term is −3-3 . So, putting a=1,n=2,an=−3a=1,n=2,{{a}_{n}}=-3 , we get,
−3=(1)r2−1 ⇒r=−3 \begin{aligned} & -3=(1){{r}^{2-1}} \\\ & \Rightarrow r=-3 \\\ \end{aligned}
To get the next three terms, we put n=5,6,7n=5,6,7 one by one in the above formula and get,
a5=(−3)5−1 ⇒a5=81 \begin{aligned} & {{a}_{5}}={{\left( -3 \right)}^{5-1}} \\\ & \Rightarrow {{a}_{5}}=81 \\\ \end{aligned}
a6=(−3)6−1 ⇒a6=−243 \begin{aligned} & {{a}_{6}}={{\left( -3 \right)}^{6-1}} \\\ & \Rightarrow {{a}_{6}}=-243 \\\ \end{aligned}
a7=(−3)7−1 ⇒a7=729 \begin{aligned} & {{a}_{7}}={{\left( -3 \right)}^{7-1}} \\\ & \Rightarrow {{a}_{7}}=729 \\\ \end{aligned}

Thus, we can conclude that the next three terms will be 81,−243,72981,-243,729 .

Note: The first mistake that we can make here is that we overlook the negative sign and directly assume the common ratio to be 33 instead of −3-3 . This will lead to partially wrong answers. So, we should be careful of it. Secondly, we should not use the wrong general expression of a geometric sequence.