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The sum of the first $3$ terms of a geometric sequence is $65$ and their product is $3375$.

How do I find the first three terms? I know that the answer is $5$, $15$ and $45$, but I don't know how to perform the steps to get the answer.

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5 Answers

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Let the terms of the sequence be $a, ar, ar^2$. Their product is $(ar)^3=3375$, so $ar=15$. Also, $a(1+r+r^2)=65$. So $\frac{1+r+r^2}{r}=\frac{65}{15}$. Multiply both sides by $3r$ and combine like terms to get $3r^2 - 10r + 3 = 0$. Solving this quadratic, $r=3$ or $\frac{1}{3}$. Using $ar=15$, these correspond to $5, 15, 45$ and $45, 15, 5$, respectively.

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First three terms are $a$, $aq$, $aq^2$. You have $$ a(1+q+q^2) =65\\ a\cdot aq\cdot aq^2 = 3375 $$ From last equation $$ (aq)^3 = 3375\Longrightarrow aq = 15 $$ Dividing first equation, we get $$ 1+q+q^2 = \frac{65}{15}q. $$ Could you proceed?

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Let the terms be $a/r, a, ar$ so that $a/r\cdot a\cdot ar=3375=15^3$

If $a$ is real, $a=15$

and consequently, $\dfrac{15}r+15+15r=65\iff3r^2-10r+3=0\implies r=?$

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Series questions at this level can typically be solved using two techniques, writing out the series using notation, and knowing the sum of an infinite series.

Any geometric series can be specified using two constants ($a_o$ and r) and is defined by the following recurrence relationship: $a_{n+1} = r * a_n$.

Another way to express a geometric series is $a_n = r^n *a_0$.

If the sum of the first three terms is 65, we have $a_0 + r a_0 + r^2 a_0 = (1+r+r^2)a_0=65$.

The second condition is simply $r^3 * a_0^3 = 3375$.

We have two conditions and two unknowns so we can apply algebra.

The second condition yields $r*a_0 = 15$ or $a_0 = 15/r$. The first condition becomes: $(1+r+r^2)(15/r) = 65$

This is an algebraic equation in one variable. Technically it's a third degree polynomial but we can guess the answer and it's 3.

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Yet another solution, assuming that it should be easy, i.e., solved in integers. $a+aq+aq^2=a(1+q+q^2)=65=5\cdot13$. Now it is easy to guess, that $a=5$ and $q=3$ is the solution of $1+q+q^2=13$ and to verify, that the condition $a^3q^3=3375$ is also fulfilled.

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