Question

The sum of three decreasing numbers in A.P. is \[27\] . If \[ - 1, - 1,3\] are added to them respectively, the resulting series is in G.P. The numbers are
A.\[5,9,13\]
B.\[15,9,3\]
C.\[13,9,5\]
D.\[17,9,1\]

Answer 1

Hint: A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series.

Complete step-by-step answer:
A geometric progression or a geometric sequence is the one, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant (which is non-zero) to the preceding term.
If \[a,b,c\] are in GP then \[{b^2} = ac\]
Let \[a + d,a,a - d\] be the three numbers in an AP.
Their sum \[ = a + d + a + a - d = 27\]
Which gives us \[3a = 27\]
And hence we get \[a = 9\]
Adding \[ - 1, - 1,3\] to these numbers respectively we get ,
\[9 + d - 1,9,9 - d + 3\] i.e. \[8 + d,8,12 - d\] are in GP.
Since we know that if \[a,b,c\] are in GP then \[{b^2} = ac\]
Hence applying this property on the terms of GP we get ,
\[{8^2} = (8 + d)(12 - d)\]
Which simplifies to
\[64 = 96 + 12d - 8d - {d^2}\]
Which simplifies to
\[{d^2} - 4d - 32 = 0\]
Which further simplifies to
\[(d - 8)(d + 4) = 0\]
Hence we get
\[d = 8\] or \[d = - 4\]
When \[d = - 4\] numbers of given AP are \[5,9,13\]( not possible because AP is decreasing)
When \[d = 8\] numbers of given AP are \[17,9,1\]
Therefore option (D) is the correct answer.
So, the correct answer is “Option D”.

Note: A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same. The behavior of the arithmetic progression depends on the common difference d. If the common difference is positive, then the members (terms) will grow towards positive infinity or negative, then the members (terms) will grow towards negative infinity.