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Marianna [84]
3 years ago
8

Please i need an answer urgently. Sum of 3 terms are 252, first term is 192. Find common ratio

Mathematics
1 answer:
Viefleur [7K]3 years ago
6 0

Answer:

r=\frac{1}{4}\\\\r=-\frac{5}{4}

Step-by-step explanation:

<u>Geometric Sequences</u>

The geometric sequence is given as:

a_1, a_1r, a_1r^2, a_1r^3,..., a_1r^{n-1}

Where n is the number of the term, n≥1, and r is the common ratio.

The sum of n terms of the geometric sequence is given by:

\displaystyle S_n=a_1\frac{r^n-1}{r-1}

We are given: S3=252, a1=192, thus substuting:

\displaystyle 252=192\frac{r^3-1}{r-1}

Dividing by 12:

\displaystyle 21=16\frac{r^3-1}{r-1}

Recall that:

r^3-1=(r-1)(r^2+r+1)

Substituting:

\displaystyle 21=16\frac{(r-1)(r^2+r+1)}{r-1}

Simplifying:

21=16(r^2+r+1)=16r^2+16r+16

Rearranging:

16r^2+16r+16-21=0

16r^2+16r-5=0

Rewriting:

16r^2-4r+20r-5=0

Factoring:

4r(4r-1)+5(4r-1)=0

(4r-1)(4r+5)=0

Solving:

r=\frac{1}{4}\\\\r=-\frac{5}{4}

Both solutions are valid

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where, \bar X = sample mean age of selected students = 22.4 years

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So, 99% confidence interval for the population​ mean, \mu is ;

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P(-2.878 < \frac{\bar X - \mu}{\frac{s}{\sqrt{n} } } < 2.878) = 0.99

P( -2.878 \times {\frac{s}{\sqrt{n} } } < {\bar X - \mu} < 2.878 \times {\frac{s}{\sqrt{n} } } ) = 0.99

P( \bar X -2.878 \times {\frac{s}{\sqrt{n} } < \mu < \bar X +2.878 \times {\frac{s}{\sqrt{n} } ) = 0.99

<u>99% confidence interval for</u> \mu = [ \bar X -2.878 \times {\frac{s}{\sqrt{n} } , \bar X +2.878 \times {\frac{s}{\sqrt{n} } ]

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Therefore, 99% confidence interval for the population​ mean is [19.891 , 24.909].

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