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15. A number in the series below has been omitted. What is the missing number?

[tex]\[
\begin{array}{llllll}
\frac{1}{288} & ? & \frac{1}{72} & \frac{1}{36} & \frac{1}{18} & \frac{1}{9}
\end{array}
\][/tex]



Answer :

GinnyAnswer
To determine the missing number in the provided series, we need to identify the pattern among the given terms.

The series provided is:
[tex]\[ \frac{1}{288}, \, ?, \, \frac{1}{72}, \, \frac{1}{36}, \, \frac{1}{18}, \, \frac{1}{9} \][/tex]

First, let's examine the sequence from the third term onward to better understand the pattern:

3. [tex]\(\frac{1}{72}\)[/tex]
4. [tex]\(\frac{1}{36}\)[/tex]
5. [tex]\(\frac{1}{18}\)[/tex]
6. [tex]\(\frac{1}{9}\)[/tex]

Notice that each term is obtained by multiplying the previous term by 2:
[tex]\[ \frac{1}{72} \times 2 = \frac{1}{36} \][/tex]
[tex]\[ \frac{1}{36} \times 2 = \frac{1}{18} \][/tex]
[tex]\[ \frac{1}{18} \times 2 = \frac{1}{9} \][/tex]

Using this pattern, we know that each term in the sequence can be derived by doubling the previous term.

Now, let's start from the first term and use this pattern to find the second term, which is the missing number:
[tex]\[ \text{First term: } \frac{1}{288} \][/tex]

If we follow the sequence, the second term should be twice the first term:
[tex]\[ \frac{1}{288} \times 2 = \frac{1}{144} \][/tex]

Therefore, the missing number in the series is:
[tex]\[ \boxed{\frac{1}{144}} \][/tex]

Converting [tex]\(\frac{1}{144}\)[/tex] to its decimal form for verification purposes, we get:
[tex]\[ \frac{1}{144} = 0.006944444444444444 \][/tex]

Thus, the missing number in the series is correctly found to be [tex]\( \boxed{\frac{1}{144}} \)[/tex] or 0.006944444444444444 in decimal form.
chibabykalu16

Answer:

1/144

Step-by-step explanation:

Given:

  • [tex]\begin{array}{llllll}\frac{1}{288} & \boxed{?} & \frac{1}{72} & \frac{1}{36} & \frac{1}{18} & \frac{1}{9}\end{array}[/tex]

To find the omitted number of the Geometric Progression (GP) we need first to find the ratio.

ratio in geometric progressions is defined as the ratio of a value against the previous value in the series e.g [tex]a_2:a_1 \text{ or } a_4:a_3[/tex]

[tex]Ratio = a_4:a_3\\ =\frac{1}{36}:\frac{1}{72}= \frac{\frac{1}{36}}{\frac{1}{72}}=\frac{1}{36} \times \frac{72}{1} = 2[/tex]

So for each proceeding term, we multiply the current term by 2

Plotting this to the formula:

[tex]a_n = a \times r^{n-1}[/tex]

Where:

  • [tex]a_n[/tex] is the value of the nth term
  • a is the first term aka [tex]a_1[/tex]
  • r represents the ratio
  • n represents the term number

[tex]a_2 = \frac{1}{288} \times 2^{2-1}[/tex]

[tex]a_2 = \frac{1}{288} \times 2^{1} =\frac{2}{288} = \frac{1}{144}[/tex]

Therefore, the omitted number is equal to 1/144.

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