Question

In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first will be
A. 2
B. 4
C. 6
D. 8

Answer 1

Hint:
If \[a,{\rm{ }}b,c.............G.P\], then use the property
\[{b^2} = ac\].
The four numbers should be:
\[a,{\rm{ }}b,{\rm{ }}c,d\]
Given that the final three numbers are in the A.P format with a common difference of\[6\],
\[ \Rightarrow b,{\rm{ }}c,{\rm{ }}d..........................A.P\]
Formula used:
If \[a,{\rm{ }}b,c.............G.P\], then use the property
\[{b^2} = ac\].
three numbers are are in AP then
\[a - d,a,a + d\]
Complete step-by-step solution:
We now understand that in A.P., the difference between the second and first numbers is equal to the difference between the third and second numbers, which is equal to the common difference, which is \[6\] as given.
That is, common difference, \[d = 6\]
Now, let us assume that the three numbers are,
\[a - d,a,a + d\]
Additionally, the first and last numbers are the same, being
\[a + d\]
The four numbers are therefore;
\[a + d,{\rm{ }}a - d,a,a + d\] -- (1)
Of the above said four, the first three are in Geometric progression.
Therefore, \[{(a - d)^2} = a(a + d)\]
But, we know that the given common difference is
\[d = 6\]
Therefore, substituting \[d = 6\] in the above equation, we get
\[{(a - 6)^2} = a(a + 6)\]
Now, we have to expand the above equation using square formula, we obtain
\[{a^2} - 12a + 36 = {a^2} + 6a\]
On grouping the like terms and after simplifying, we get
\[18a = 36\;\]
Now, solve for\[a\]by dividing either side of the equation by \[18\]:
\[a = 2\]
Now, we have to find all the four terms of the G.P:
Here, first term according to (1) is
\[ = a + d\]
\[ = 2 + 6\]
\[ = 8\]
Second term according to (1) is
\[ = a - d\]
\[ = 2 - 6\]
\[ = - 4\]
Third term according to (1) is
\[ = a\]
\[ = 2\]
Fourth term according to (1) is
\[ = a + d\]
\[ = 2 + 6\]
\[ = 8\]
Therefore, the required series is \[8, - 4,2,8\]
Hence, the option D is correct.
NOTE:
The most important thing to keep in mind when answering these types of questions is to always keep in mind the above-mentioned properties of the arithmetic and geometric progressions. By applying these properties and meeting the specified requirements, we will be able to determine the correct response.