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Chapter 22: Problem 683

The seventh term of a geometric progression is 192 and \(\mathrm{r}=2\). Find the first four terms.

Short Answer

Expert verified
The first four terms of the geometric progression are 3, 6, 12, and 24.

Step by step solution

01

Use the geometric progression formula to find the first term.

The formula for the nth term of a geometric progression is given by: \[T_n = ar^{n-1}\] Where \(T_n\) is the nth term, \(a\) is the first term, and \(r\) is the common ratio. We are given that \(T_7 = 192\) and \(r = 2\). Plugging these values into the formula, we get: \[192 = a(2)^{7-1}\]
02

Solve the equation for the first term.

To find the value of \(a\), we must solve the equation that we derived in Step 1. \[192 = a(2)^6\] Divide both sides of the equation by \(2^6\), which is 64: \[a = \frac{192}{64}\] Simplify the fraction to get the first term: \[a = 3\]
03

Use the first term to find the remaining terms.

Now that we have the first term, we can find the remaining terms in the geometric progression by multiplying the previous term by the common ratio \(r\). The first term, \(a\), is 3. Multiply this term by the common ratio (2) to get the second term: \[T_2 = 3 \times 2 = 6\] Multiply the second term by the common ratio to get the third term: \[T_3 = 6 \times 2 = 12\] Multiply the third term by the common ratio to get the fourth term: \[T_4 = 12 \times 2 = 24\]
04

Write down the first four terms of the geometric progression.

We have now found the first four terms of the geometric progression, which are: \[T_1 = 3\] \[T_2 = 6\] \[T_3 = 12\] \[T_4 = 24\] Therefore, the first four terms of the geometric progression are 3, 6, 12, and 24.

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Most popular questions from this chapter

Insert 20 arithmetic means between 4 and \(67 .\)

Find the sum of the first ten terms of the geometric progression: \(15,30,60,120\)

Find the twelfth term of the arithmetic sequence \(2,5,8, \ldots\).

Find three numbers in geometric progression whose sum is 19, and whose product is 216 .

If \(\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}, \ldots \mathrm{s}_{\mathrm{p}}\) are the sums of \(\mathrm{n}\) terms of an arithmetic progression whose first terms are \(1,2,3,4, \ldots\) and whose common differences are \(1,3,5,7, \ldots\) respectively, find the value of \(\quad s_{1}+s_{2}+s_{3}+\ldots s_{p}\)
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