Question

If \[{a_1},{\text{ }}{a_2},...{a_{50}}\;\] are in G.P. then \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}}\] is equal to:
A. \[0\]
B. \[1\]
C. \[\dfrac{{{a_1}}}{{{a_2}}}\]
D. \[\dfrac{{{a_{25}}}}{{{a_{24}}}}\]

Answer 1

Hint: In this question, we need to find the value of \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}}\] .
For this, we need to use the concept of geometric sequence to get the desired result.

Complete step-by-step answer:
We know that the given sequence is \[{a_1},{\text{ }}{a_2},...{a_{50}}\;\]
It is in the form of \[a,{\text{ }}ar,...a{r^{n - 1}}\;\]
So, first term of a sequence is \[{a_1} = ar\;\]
Like this, we get
\[{a_2} = a{r^2}\;\]
\[{a_3} = a{r^3}\;\]
\[{a_n} = a{r^{n - 1}}\;\]and so on.
Also, \[r\] is the common ratio of the sequence.
Thus, we get
 \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}} = \dfrac{{(a - a{r^2} + a{r^4} + \ldots a{r^{48}})}}{{(ar - a{r^3} + a{r^5} + \ldots a{r^{49}})}}\]
By taking r common from the denominator, we get
\[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}} = \dfrac{{(a - a{r^2} + a{r^4} + \ldots a{r^{48}})}}{{r(a - a{r^2} + a{r^4} + \ldots a{r^{48}})}}\]
By simplifying further, we get
\[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}} = \dfrac{1}{r}\]
That means, \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}} = \dfrac{a}{{ar}}\]
So, we can say that \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}} = \dfrac{{{a_1}}}{{{a_2}}}\]
Hence, the value of \[\dfrac{{\left( {{a_1} - {a_3} + {a_5} - ... + {a_{49}}} \right)}}{{\left( {{a_2} - {a_4} + {a_6} - ... + {a_{50}}} \right)}}\] is \[\dfrac{{{a_1}}}{{{a_2}}}\].
Therefore, the correct option is (C).

Additional Information: Geometric Progression or Geometric Series A G.P. is created by multiplying every number or part of a series by the very same number. This is known as the constant ratio. The ratio of any two consecutive numbers in a G.P. is just the same number, which we consider the constant ratio. We can say that if any three numbers are in geometric progression series, then the middle number is the geometric mean of the remaining two numbers. Geometric progressions can be either finite or infinite. Its common ratio is either negative or positive.

Note: Here, the geometric sequence plays an important role in solving this question. It is necessary to compare the coefficients of a given sequence with the standard geometric sequence to find the first, second terms and so on. The given geometric sequence is a finite sequence.