Question

If \[a,b,c\]are in A.P., then \[{10^{ax + 10}},{10^{bx + 10}},{10^{cx + 10}}\]will be in
A. A.P.
B. G.P only when \[x > 0\]
C. G.P for all values of \[x\]
D. G.P for \[x < 0\]

Answer 1

Hint
A series of terms known as an arithmetic progression (AP) have identical differences. In a geometric progression (GP), the common ratio is multiplied by the previous term to produce each succeeding term. With the aid of a shared distinction between the two following terms, the series is recognized in AP. Geometric Progression: To create new series, multiply two successive terms so that their factors are constant. A common ratio between consecutive terms in GP is used to identify the series.
to figure out how many spectators a stadium can accommodate. The real-world example is when we take a cab, we will be charged an initial fee and then a fee per mile or kilometre.
Formula use:
The arithmetic progression is \[a,b,c\]
\[(b - a) = (c - b)\]
The geometric progression is \[a,b,c\]
\[{b^2} = ac\].
Complete step-by-step solution
The AP series progression is \[a,b,c\]
\[b - a = c - b\]
The equation can be written as
\[ = > bx - ax = cx - bx\]
\[ = > (bx + 10) - (ax + 10) = (cx + 10) - (bx + 10)\]
\[{10^{(bx + 10) - (ax + 10)}} = {10^{(cx + 10) - (bx + 10)}}\]
This equation is equal to
\[ = > {10^{(bx + 10)}} = \sqrt {({{10}^{an + 10}})({{10}^{bx + 10}})} \]
So, \[{10^{ax + 10}},{10^{bx + 10}},{10^{cx + 10}}\]are involved in Geometric progression.
Therefore, the correct option is C.

Note
The AP cost per invoice is calculated by dividing the total number of invoices paid during a specified period of time by the total costs incurred to pay those invoices. This indicator offers a precise assessment of a company's AP efficiency, together with other accounts payable KPIs. Geometric progression is the name given to a sequence in which the ratio of two succeeding words is constant.
 The first n terms of the arithmetic sequence are added together to form the sum of the first n terms of AP. There is a fixed ratio between every pair of subsequent terms in a geometric series. This would produce a constant multiplier effect.