Question

Which term of the G.P. $4,-8,16,-32......$ is $1024$?

Answer 1

Firstly, we need to find the “first term” from the given geometric series. Then we shall proceed to calculate the common ratio from the given geometric series. Lastly, put the values of first term and common ratio calculated, in the formula used to find the ”nth” term of the geometric series to find the answer.

Complete answer:
Before proceeding to the solution, we shall gather some basic information about G.P. (Geometric Progression).
In MATHEMATICS, “geometric progression” is also known as “Geometric Series”. Geometric series is a sequence of non zero numbers. This series is formed by multiplying its first term by a common ratio in order to form the next number. The same process is carried forward to form the series.
First term $\left( a \right)$ is the first and foremost term of the series that is multiplied by a common ratio in order to form the next term of the series. The next term is then again multiplied by the same common ratio to discover the next term to it. The process is continued to form the series till “FINITE” or “INFINITE” terms.
Common ratio $\left( r \right)$ is termed as the only number (In a particular series) which is multiplied by the previous term in order to discover the next term to it.
In other words, we can say that it is the number formed by dividing a term in a series by the term just before the same term.
The common ratio remains the same for each term in the entire series for a particular series.
Let us see the expression of geometric series,
$a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},......,a{{r}^{n-1}}$
Here,
$a$ is the first term which is further multiplied by common ratio $r$ to form the next term of the series, i.e., $ar$ and this process is repeated to form other terms of the same series.
Now, let us proceed to the solution.
Let $1024$ be the ${{n}^{th}}$ term of the geometric series $4,-8,16,-32......$.
Here,

& a=4 \\\ & r=\dfrac{-8}{4}=-2 \\\ & \\\ \end{aligned}$$ $${{a}_{n}}=1024$$ We now need to put the values above in the equation of the $${{n}^{th}}$$ term of a G.P. $$a\times {{\left( r \right)}^{n-1}}={{a}_{n}}$$ On putting the values, we get, $$\begin{aligned} & 4\times {{\left( -2 \right)}^{n-1}}=1024 \\\ & {{\left( -2 \right)}^{n-1}}=256 \\\ & {{\left( -2 \right)}^{n-1}}={{\left( -2 \right)}^{8}} \\\ & \\\ \end{aligned}$$ On comparing we get, $$\begin{aligned} & n-1=8 \\\ & n=9 \\\ \end{aligned}$$ So, $$1024$$ is the $${{9}^{th}}$$ term of the given series. **The final answer is $${{9}^{th}}$$ term.** **Note:** Other than “GEOMETRIC PROGRESSIONS”, two other types of series also exist, namely “ARITHMETIC PROGRESSIONS” and “HARMONIC PROGRESSIONS”. “HARMONIC PROGRESSIONS” are the progressions formed by taking the reciprocals of “ARITHMETIC PROGRESSIONS”. If number of terms in any AP, GP or HP is ”odd”, then the AM (ARITHMETIC MEAN), GM(GEOMETRIC MEAN) and HM(HARMONIC MEAN), respectively, of the first term and the last term is the middle term of the series. Students often encounter confusions while calculating the “nth” term of the series. The simplest formula to keep in mind is, $${a_n} = a \times {r^{\left( {n - 1} \right)}}$$ where $$n \to $$ $number\;of\;terms\;in\;the\;series$