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Step-by-Step Solution
Step 1: Represent the three terms of the G.P.
Let the three consecutive terms of the geometric progression (G.P.) be $ \frac{a}{r}, \, a, \, ar $. These are consecutive because each term is obtained by multiplying the previous one by the common ratio $r$.
Step 2: Use the given product condition
The problem states that the product of the three G.P. terms is 512. Mathematically, that means:
$
\left(\frac{a}{r}\right) \times a \times (ar) = a^3 = 512
$
Solving $ a^3 = 512 $ gives $ a = 8 $ because $ 512 = 8^3 $.
Step 3: Form the new A.P. after adding 4 to the first two terms
When 4 is added to each of the first two terms, the new three terms become:
$ \left(\frac{a}{r} + 4\right), \quad (a + 4), \quad (ar).
$
But it is given that these three terms form an arithmetic progression (A.P.). In an A.P., the difference between consecutive terms is constant. Thus,
$ (a + 4) - \left(\frac{a}{r} + 4\right) = (ar) - (a + 4).
$
Step 4: Substitute $ a = 8 $ and simplify
Substituting $ a = 8 $ in the above equation, we get:
$ (8 + 4) - \left(\frac{8}{r} + 4\right) = (8r) - (8 + 4).
$
Simplify each side:
Left side: $ 12 - \left(\frac{8}{r} + 4\right) = 12 - \frac{8}{r} - 4 = 8 - \frac{8}{r}. $
Right side: $ 8r - 12.
$
So the equation becomes:
$ 8 - \frac{8}{r} = 8r - 12.
$
Step 5: Solve for $ r $
Rearrange the equation to one side:
$ 8 - \frac{8}{r} - 8r + 12 = 0
$
$ 20 - 8r - \frac{8}{r} = 0.
$
Multiply through by $ r $ to clear the fraction:
$ 20r - 8r^2 - 8 = 0.
$
Rewrite it:
$ -8r^2 + 20r - 8 = 0.
$
Divide through by $ -4 $ to simplify:
$ 2r^2 - 5r + 2 = 0.
$
Factorize or use the quadratic formula. Factorization gives:
$ (2r - 1)(r - 2) = 0,
$
which yields the solutions:
$ r = \frac{1}{2} \quad \text{or} \quad r = 2.
$
Step 6: Find the sum of the original G.P. terms
The original three terms are:
$ \frac{a}{r}, \quad a, \quad ar.
$
With $ a = 8 $, the terms become:
If $ r = 2 $, the terms are $ \frac{8}{2} = 4, \, 8, \, 16 $. Their sum is $ 4 + 8 + 16 = 28 $.
If $ r = \frac{1}{2} $, the terms are $ \frac{8}{\frac{1}{2}} = 16, \, 8, \, 4 $. Their sum is $ 16 + 8 + 4 = 28 $.
In either case, the sum of the three original G.P. terms is $ 28 $.
Final Answer
The sum of the original three terms is 28.
