Question

Which term of GP: 3 -6 12 -24…. is -384?

Answer 1

Hint: GP mentioned in the question stands for Geometric Progression. This refers to a series of numbers where the terms to succeed are given by the product of the preceding terms with a constant value called common ratio.

Complete step-by-step answer:
Given GP: 3 -6 12 -24….
First term (a) = 3
Common ratio (r) = $ - \dfrac{6}{3} = - 2$
${n^{th}}$ term ( ${a_n}$ ) = -384
Substituting these values in the formula of ${n^{th}}$term of GP, we get:
$
  {a_n} = a{r^{n - 1}} \\
   - 384 = 3 \times {( - 2)^{n - 1}} \\
  \dfrac{{ - 384}}{3} = {( - 2)^{n - 1}} \\
   - 128 = {( - 2)^{n - 1}} \\
  {( - 2)^7} = {( - 2)^{n - 1}} \\
 $ [To get the like terms]
Now,
n – 1 = 7 [As both of these are the powers of (-2)]
n = 7 + 1
n = 8
Therefore, -384 is ${8^{th}}$ term of the given GP.

Note: The powers of like terms are equal.
Common ratio of a GP can be calculated by dividing any two terms (succeeding as numerator, preceding as denominator) as it is always constant.
The general form of a GP is:
${a_1}r,{a_2}r,{a_3}r.......{a_n}r$
And in terms of only one value (as r is constant), it can be written as:
${a_1}r,{a_1}{r^2},{a_1}{r^3}......$
The reciprocals of all the terms in a GP are also in GP.