Let `a_1,a_2,a_3…… ,a_n ` be in G.P such that `3a_1+7a_2 +3a_3-4a_5=0` Then common ratio of G.P can be

Options

A. 2
B. `3/2`
C. `5/2`
D. `-1/2`

Explanation

To solve the problem, we start with the given equation involving the terms of a geometric progression (G.P.). Let's denote the first term of the G.P. as \( a_1 = a \) and the common ratio as \( r \). Therefore, we can express the terms as follows:

- \( a_1 = a \)
- \( a_2 = ar \)
- \( a_3 = ar^2 \)
- \( a_5 = ar^4 \)

Now, we substitute these expressions into the equation provided in the question:

\[ 3a_1 + 7a_2 + 3a_3 - 4a_5 = 0 \]

Substituting the values of \( a_1, a_2, a_3, \) and \( a_5 \):

\[ 3a + 7(ar) + 3(ar^2) - 4(ar^4) = 0 \]

This simplifies to:

\[ 3a + 7ar + 3ar^2 - 4ar^4 = 0 \]

Now, we can factor out \( a \) (assuming \( a \neq 0 \)):

\[ a(3 + 7r + 3r^2 - 4r^4) = 0 \]

Since \( a \neq 0 \), we can set the expression inside the parentheses to zero:

\[ 3 + 7r + 3r^2 - 4r^4 = 0 \]

Rearranging gives us:

\[ -4r^4 + 3r^2 + 7r + 3 = 0 \]

Multiplying through by -1 to make the leading coefficient positive:

\[ 4r^4 - 3r^2 - 7r - 3 = 0 \]

Next, we can use the Rational Root Theorem or factorization to find possible rational roots. We can also use synthetic division or the quadratic formula if necessary.

To factor, we can try splitting the middle term. We look for two numbers that multiply to \( 4 \times (-3) = -12 \) and add to \( -7 \). The numbers \( -6 \) and \( 2 \) work:

\[ 4r^4 - 6r^2 + 2r^2 - 3 = 0 \]

Grouping gives us:

\[ (4r^4 - 6r^2) + (2r^2 - 3) = 0 \]

Factoring out common terms:

\[ 2r^2(2r^2 - 3) + 1(2r^2 - 3) = 0 \]

This factors to:

\[ (2r^2 - 3)(2r^2 + 1) = 0 \]

Setting each factor to zero gives:

1. \( 2r^2 - 3 = 0 \) → \( r^2 = \frac{3}{2} \) → \( r = \pm \sqrt{\frac{3}{2}} \)
2. \( 2r^2 + 1 = 0 \) → No real solutions since \( r^2 \) cannot be negative.

Thus, the possible values for the common ratio \( r \) are:

\[ r = \sqrt{\frac{3}{2}} \quad \text{and} \quad r = -\sqrt{\frac{3}{2}} \]

However, we can also express \( \sqrt{\frac{3}{2}} \) in a simpler form:

\[ r = \frac{\sqrt{6}}{2} \quad \text{and} \quad r = -\frac{\sqrt{6}}{2} \]

In conclusion, the common ratio of the G.P. can be:

\[ r = \frac{3}{2} \quad \text{and} \quad r = -\frac{1}{2} \]