Question

What is a possible value for the missing term of the geometric sequence 1250, __, 50, … ?

Answer 1

Hint: We know that if any three terms $a,b,c$ are in geometric sequence, then the middle term, that is, b is called the Geometric Mean of a and c. Also, we are very well aware that G.M. = b = $\sqrt{ac}$. Using this concept, we can find the value of the missing term.

Complete step-by-step solution:
We know that, in a sequence ${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}},...$ if the ratio of consecutive terms is same, that is, if $\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{3}}}{{{a}_{2}}}=\dfrac{{{a}_{4}}}{{{a}_{3}}}=...=r$, then such a sequence is called a geometric sequence. Here, ${{a}_{1}}$ is called the first term and r is called the common ratio of this geometric sequence.
Now, let us assume a geometric sequence $a,b,c$.
Since this is a geometric sequence, we know that the ratio of consecutive terms will be constant.
Thus, we have $\dfrac{b}{a}=\dfrac{c}{b}...\left( i \right)$.
Let us rearrange the terms in equation (i) to get
${{b}^{2}}=ac$
Or, we may write this as,
$b=\sqrt{ac}$
We must remember that b is also the Geometric Mean (G.M.) of this sequence.
Thus G.M. = b = $\sqrt{ac}$.
Here, in our question, we are given that 1250, __, 50 are in geometric sequence.
Let the missing term be $x$. So now, we can say that 1250, $x$, 50 are in a geometric sequence.
So, by using the concept of Geometric Mean, we can say that $x$ will be the Geometric Mean of this sequence.
So now, we have
$x=\sqrt{1250\times 50}$
We know that the prime factorization of $1250=2\times 5\times 5\times 5\times 5$ and that of $50=2\times 5\times 5$.
Thus, we have $x=2\times 5\times 5\times 5$.
Or, $x=250$.
Hence, the missing term in the given geometric sequence is 250.

Note: We must remember the difference between a sequence and a series. We know that a sequence is a collection of related elements or terms, whereas, a series is defined as the sum of elements of any sequence.