Question

How do you find the explicit formula for the following sequence \[\dfrac{1}{2},\dfrac{3}{4},\dfrac{5}{8},\dfrac{7}{{16}}\] . . .?

Answer 1

Hint: As we can see that in given sequence the numerator terms form arithmetic sequence as there is a common difference between each term and denominator terms form geometric sequence, hence by applying the conditions of both AP and GP we can find the formula of the given sequence.

Formula used:
\[{a_n} = {a_1} + \left( {n - 1} \right)d\]
\[{a_n}\]is the nth term
\[{a_1}\] is the first term
\[n\] is the terms number in the sequence
\[d\] is the common difference
\[{q_n} = a \cdot {r^{n - 1}}\]
\[a\]is the first term
\[r\]is the common ratio.

Complete step by step solution:
In the given sequence the numerator term forms arithmetic sequence: 1, 3, 5, 7
As there is a common difference between each term. In this case, adding 2 to the previous term of numerator in the sequence gives the next term.
An explicit formula of an arithmetic series includes all given information as
\[\Rightarrow {p_n} = {a_1} + \left( {n - 1} \right)d\]
This is the formula of an arithmetic sequence.
In the given sequence with respect to the numerators term, we get d as
\[\Rightarrow d = {a_2} - {a_1}\]
\[\Rightarrow d = 3 - 1 = 2\]
Substitute in the values of $a_1$=1 and d=2 to get the formula for numerator terms
\[\Rightarrow {p_n} = {a_1} + \left( {n - 1} \right)d\]
\[\Rightarrow {p_n} = 1 + \left( {n - 1} \right)2\]
To simplify each term, apply distributive property
\[\Rightarrow {p_n} = 1 + 2n - 2\]
Therefore, we get
\[\Rightarrow {p_n} = 2n - 1\]
In the given sequence the denominator term forms a geometric sequence: 2, 4, 8, 16 with common ratio as 2.
An explicit formula of a geometric series includes all given information as
\[\Rightarrow {q_n} = a \cdot {r^{n - 1}}\]
Substitute in the values of a=2 and r=2 to get the formula for denominator terms
\[\Rightarrow {q_n} = 2 \cdot {2^{n - 1}}\]
Therefore, we get
\[\Rightarrow {q_n} = {2^n}\]
Thus, the explicit formula for the following sequence is given as
\[\Rightarrow {a_n} = \dfrac{{{p_n}}}{{{q_n}}}\]
Substituting the values of \[{p_n}\] and \[{q_n}\]we get
\[\Rightarrow {a_n} = \dfrac{{2n - 1}}{{{2^n}}}\]
Hence, by applying this formula we can get the following sequence.

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.