First slide

Geometric progression

Question

The sum of three numbers in G.P. is 14. If one is added to the first and second numbers and I is subtracted from the third, the new numbers are in A.P. The smallest of them is

Moderate

A

2
B

4
C

6
D

10

Solution

Let the numbers be a, ar, ar². Then,
a + ar + ar² = 14 (given)----------(1)
Now,
a + 1 ar + l, ar²- 1 are in A.p.
$\begin{array}{l} \Rightarrow 2 (ar + 1) = a + 1 + {ar}^{2} - 1 \\ \Rightarrow 2 ar + 2 = a + {ar}^{2} - - - - (2) \end{array}$
From (1) and (2),
2ar+2=14-ar
or 3ar= 12
or ar=4 --------(3)
From (1),
a+4+4r=L4
a+4r=10 --------(4)
From (3) and (4),
$a + \frac{16}{a} = 10 \Rightarrow a = 2, 8$
Hence, the smallest number is 2.

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