Question

If all the terms of an A.P. are squared, then new series will be in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint
Each term in an arithmetic-geometric progression (AGP) can be modeled as the union of the terms from an arithmetic progression (AP) and a geometric progression (GP) (GP). An arithmetic progression is a set of integers with a fixed difference between any two succeeding numbers (A.P.).
Arithmetic Progressions' Characteristics are
The terms in the sequence that result if the same number is added to or removed from each term of an A.P. are also in an A.P. with the same common difference. The resulting sequence is also an A.P if each word in an A.P is divided or multiplied by the same non-zero value.
Formula used:
In AP constant difference between each term and the term before it.
AP sequence is
\[a,a + d,a + 2d,a + 3d\]
Complete step-by-step solution
Any series is said to as an arithmetic progression if there is a constant difference between each term and the term before it.
Assume that AP sequence is
\[a,a + d,a + 2d,a + 3d\]
Square all the terms of the sequence,
\[ = > {a^2},({a^2} + 2ad + {d^2}),({a^2} + 4ad + 4{d^2}),({a^2} + 6ad + 9{d^2})....\]
It is obvious that the series does not meet any requirements for an AP, GP, or HP.
So, if all the terms of an A.P. are squared, then the new series is not the either series.
Therefore, the correct option is D.

Note
Any series is said to as an arithmetic progression if there is a constant difference between each term and the term before it. Any non-zero numerical sequence is referred to be a geometric progression if the ratio between each term and the term before it is consistently constant. If the reciprocal of a sequence of non-zero numbers, such as \[{a_1},{\rm{ }}{a_2},{\rm{ }}{a_3},...{a_n},\]is itself a sequence, the progression is said to be harmonic.