Question

In a set of 4 numbers first three are in G.P. and last three numbers are in A.P. with a common difference of 6. If the first number is the same as the 4th number. Find the four numbers.

Answer 1

Hint: 1) G.P. stands for Geometric Progression. It is a special form of sequence of numbers, where any given term after the first term can be found by multiplying the previous term by a fixed, non-zero number called the Common ratio. We denote first term by ‘a’ and common ratio by ‘r’
2) A.P. stands for Arithmetic Progressions. It is a special form of sequence of numbers, where any given term after the first term can be found by adding a constant term (called Common Difference) to the previous term. We denote first term by ‘a’ and common difference by ‘d’

Complete step-by-step answer:
Since, the last three numbers are in A.P. and therefore we may take the last three numbers as \[\left( {{\text{ }}a{\text{ }}-{\text{ }}d{\text{ }}} \right),{\text{ }}a,{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}d{\text{ }}} \right)\].
And since the first term and fourth term are the same. Therefore the 1st term will also be ( a + d) .
Therefore the given set of numbers will be as \[\left( {{\text{ }}a{\text{ }} + {\text{ }}d{\text{ }}} \right),{\text{ }}\left( {{\text{ }}a{\text{ }}-{\text{ }}d{\text{ }}} \right),{\text{ }}a,{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}d{\text{ }}} \right).\;\]
And since the first three numbers are in G.P.
\[\therefore \;{\left( {a{\text{ }}-{\text{ }}d} \right)^2} = {\text{ }}a\left( {a{\text{ }} + {\text{ }}d} \right)\]
And since given \[d{\text{ }} = {\text{ }}6\]
$\therefore $ \[{\left( {a{\text{ }}-{\text{ }}6} \right)^2} = {\text{ }}a\left( {a{\text{ }} + {\text{ }}6} \right)\]
\[{a^2}\; - {\text{ }}12a{\text{ }} + {\text{ }}36{\text{ }} = {\text{ }}{a^2} + {\text{ }}6a\]
\[18a{\text{ }} = {\text{ }}36\]
\[a{\text{ }} = {\text{ }}2\]
Putting the value of a and d,

The four numbers are 8, -4, 2, 8

Note: In any A.P. we take three consecutive numbers as \[\left( {{\text{ }}a{\text{ }}-{\text{ }}d{\text{ }}} \right),{\text{ }}a,{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}d{\text{ }}} \right)\]to make calculation easy. And also in any G.P. if a, b, c are three consecutive numbers then \[{b^2} = {\text{ }}a*c.\]