The sum of the first three-term of a G.P is $\dfrac{39}{10}$ and their product is 1.Find the value of the number.
Answer
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Hint: In order to solve this question regarding the geometric progression of a series, we consider the three terms of the GP as $\dfrac{a}{r}, a, ar.$ and follow the steps as given in the question to get to our final result.
Complete step by step solution: A geometric progression(sequence) (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. The geometric progression can be written as: $\text{a}\text{r}^{0}=\text{a},\;\text{a}\text{r}^{1}=\text{ar},\;\text{a}\text{r}^{2},\; \text{a}\text{r}^{3},....$ where r ≠0, r is the common ratio and a is a scale factor(also the first term). Let the first three-term of Geometric Progression be:- $\dfrac{a}{r}$, a, ar Then, according to the question; the sum of the first three terms is $=\dfrac{39}{10}.$.........given. So,$\dfrac{a}{r}+\text{a}+\text{ar}= \dfrac{39}{10}$ ……...(i) And the product of it's first three terms is 1, so we get; $\Rightarrow \dfrac{a}{r}\times a\times ar=1$ $\Rightarrow {{a}^{3}}=1={{1}^{3}}$ $\Rightarrow ;\ \text{putting}\ $ in equation (i), we get; $\Rightarrow a\left[ \dfrac{1}{r}+1+r \right]=\dfrac{39}{10}$ $\Rightarrow 1\left[ \dfrac{1}{r}+1+r \right]=\dfrac{39}{10}$ $\Rightarrow \left( \dfrac{1+r+{{r}^{2}}}{r} \right)=\dfrac{39}{10}$
$\Rightarrow 10\left( 1+r+{{r}^{2}} \right)=39\left( r \right)$ $\Rightarrow 10+10r+10{{r}^{2}}=39r$ $\Rightarrow 10+10r-39r+10{{r}^{2}}=0$ $\Rightarrow 10{{r}^{2}}-29r+10=0$
three term $=\dfrac{5}{2},1,\dfrac{2}{5}$ and for $a=1,\ \ \text{and}\ \ \ \text{r}\ \text{=}\ {5}/{2;}\;$three term $={}^{2}/{}_{5},1,{}^{5}/{}_{2}$
Note: A geometric progression(sequence) (also inaccurately known as a geometric series) is a sequence of numbers, such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. The first three terms of the G.P can be represented as:-$\dfrac{a}{r},a,ar.$ The first five terms of the G.P can be represented as :- $\dfrac{a}{{{r}^{2}}},\dfrac{a}{r},a,ar,a{{r}^{2}}.$
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