Question

In a geometric progression, if the ratio of the sum of 5 terms to the sum of their reciprocals is 49 and the sum of the first and the third terms is 35. Then the first term of this geometric progression is:
A.21
B.7
C.42
D.28

Answer 1

Hint: In order to solve the given problem first consider the 5 geometric progression terms with the help of some unknown variables as the general geometric progression terms. And then with the help of given statements in the question try to find some algebraic equation and finally by solving the equation find the first term.

Complete step by step answer:
Let us consider 5 terms in geometric progression are
$\dfrac{a}{{{r^2}}},\dfrac{a}{r},a,ar,a{r^2}$
Here r is the common difference.
In order to solve the problem we need to find the value of “a” and “r”.
So the reciprocal of these 5 terms in geometric progression will be given as:
$\dfrac{{{r^2}}}{a},\dfrac{r}{a},\dfrac{1}{a},\dfrac{1}{{ar}},\dfrac{1}{{a{r^2}}}$
Let the sum of the terms be ${S_1}$
So we have:
$
  {S_1} = \dfrac{a}{{{r^2}}} + \dfrac{a}{r} + a + ar + a{r^2} \\
   \Rightarrow {S_1} = a\left[ {\dfrac{1}{{{r^2}}} + \dfrac{1}{r} + 1 + r + {r^2}} \right] \\
 $
Let the sum of reciprocal terms is given as ${S_2}$
So, we have the sum of reciprocal terms are:
$
  {S_2} = \dfrac{{{r^2}}}{a} + \dfrac{r}{a} + \dfrac{1}{a} + \dfrac{1}{{ar}} + \dfrac{1}{{a{r^2}}} \\
   \Rightarrow {S_2} = \dfrac{1}{a}\left[ {{r^2} + r + 1 + \dfrac{1}{r} + \dfrac{1}{{{r^2}}}} \right] \\
 $
As we know that the ratio of sum of 5 terms to the sum of their reciprocals is 49. So, we have:
$ \Rightarrow \dfrac{{{S_1}}}{{{S_2}}} = 49$
Substituting the values we get:
$
  \because \dfrac{{{S_1}}}{{{S_2}}} = 49 \\
   \Rightarrow \dfrac{{{S_1}}}{{{S_2}}} = \dfrac{{\left( {a\left[ {\dfrac{1}{{{r^2}}} + \dfrac{1}{r} + 1 + r + {r^2}} \right]} \right)}}{{\left( {\dfrac{1}{a}\left[ {{r^2} + r + 1 + \dfrac{1}{r} + \dfrac{1}{{{r^2}}}} \right]} \right)}} = 49 \\
   \Rightarrow {a^2} = 49 \\
   \Rightarrow a = 7 \\
 $
From the above equation we have $a = 7$
Also, we know that the sum of the first and third term is 35. So, we have
$
   \Rightarrow {T_1} + {T_3} = 35 \\
   \Rightarrow \dfrac{a}{{{r^2}}} + a = 35 \\
 $
Let us substitute the value of “a” from the above equation.
$
   \Rightarrow \dfrac{7}{{{r^2}}} + 7 = 35 \\
   \Rightarrow \dfrac{7}{{{r^2}}} = 28 \\
   \Rightarrow {r^2} = \dfrac{7}{{28}} = \dfrac{1}{4} \\
   \Rightarrow r = \dfrac{1}{2} \\
 $
So, now we have the value of r is $\dfrac{1}{2}$
Now, let us substitute the value of “a” and “r” to find the first term. So, we have:
$
   \Rightarrow {T_1} = \dfrac{a}{{{r^2}}} = \dfrac{7}{{{{\left( {\dfrac{1}{2}} \right)}^2}}} \\
   \Rightarrow {T_1} = \dfrac{7}{{\left( {\dfrac{1}{4}} \right)}} = 7 \times 4 = 28 \\
 $
Hence, the first term of this geometric progression is 28
So, option D is the correct answer.

Note: In order to solve such problems students must remember the general terms of some common series like arithmetic progression series and geometric progression series. In geometric progression series the every next term in the series is obtained by multiplying the previous term in the series by some common term called common ratio.