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Question: How do you find the first 4 terms of a geometric sequence with a first term of 2 and a common ratio ...

How do you find the first 4 terms of a geometric sequence with a first term of 2 and a common ratio of 4?

Explanation

Solution

A geometric progression (GP) is a sequence of numbers in which each succeeding number is obtained multiplying a specific number called common ratio. We can find first 4 terms of a geometric sequence by applying the formula of nth term as an=a⋅rn−1{a_n} = a \cdot {r^{n - 1}}, hence by substituting a as 2 and r as 4 with n values up to 4 we can find the sequence.

Formula used:
qn=a⋅rn−1{q_n} = a \cdot {r^{n - 1}}
aais the first term
rris the common ratio.

Complete step by step solution:
To find the geometric sequence as per the given data we know that the first term a is 2 and common ratio r is 4.
An explicit formula of a geometric series for nth term is given as
an=a⋅rn−1{a_n} = a \cdot {r^{n - 1}} …………………. 1
As we know the first term is given as 2 i.e.,
a1=2{a_1} = 2
Hence, we need to find the next four terms of the sequence by using equation 1 as
an=a⋅rn−1{a_n} = a \cdot {r^{n - 1}}
For second term,
a2=a⋅r2−1{a_2} = a \cdot {r^{2 - 1}}
a2=2⋅41{a_2} = 2 \cdot {4^1}
\Rightarrow $$$${a_2} = 8
For third term,
a3=a⋅r3−1{a_3} = a \cdot {r^{3 - 1}}
a3=2⋅42{a_3} = 2 \cdot {4^2}
\Rightarrow $$$${a_3} = 32
For fourth term,
a4=a⋅r4−1{a_4} = a \cdot {r^{4 - 1}}
a4=2⋅43{a_4} = 2 \cdot {4^3}
\Rightarrow $$$${a_4} = 128
For fifth term,
a5=a⋅r5−1{a_5} = a \cdot {r^{5 - 1}}
a5=2⋅44{a_5} = 2 \cdot {4^4}
\Rightarrow $$$${a_5} = 512
Therefore, the first 4 terms of a geometric sequence are 2, 8, 32, 128.

Additional information:
An arithmetic progression (AP) is a sequence of numbers in which each succeeding number is obtained either by adding or subtracting a specific number called common difference. The general form of AP is: a, a + d, a + 2d, ….
A sequence of numbers is said to be a harmonic progression if the reciprocal of those numbers is in AP. Suppose A, G, H are the means of AP, GP and HP, respectively, then:
G2=AH{G^2} = AH

Note: In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, there is a constant difference between consecutive terms, the sequence is said to be an arithmetic sequence, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term.