If a constant is added to each term of an A.P. the resulting sequence is also an ______

Arithmetic Progression

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Non - Arithmetic Progression

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Fibonacci Sequence

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None of the above

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Solution

The correct option is A Arithmetic Progression
If a constant is added to each term of an A.P. the resulting sequence is also an arithmetic progression.

For example:- Let ${a_{1}, a_{2}, a_{3} \dots}$ be an arithmetic progression with common difference $d$ .
$d = a_{2} - a_{1} = a_{3} - a_{2}$

Now, $k$ be a fixed constant added to each term of the sequence so, the resulting sequence is $a_{1} + k, a_{2} + k, a_{3} + k, \dots$ ,

$\begin{matrix} d^{'} & = a_{2} + k - (a_{1} + k) = a_{2} + k - a_{1} - k = a_{2} - a_{1} = d \end{matrix}$

Similarly,
$\begin{matrix} d^{'} & = a_{3} + k - (a_{2} + k) = a_{3} + k - a_{2} - k = a_{3} - a_{2} = d \end{matrix}$

So, in the new resulting sequence difference between the consecutive terms remains the same which implies that the sequence ${a_{1} + k, a_{2} + k, a_{3} + k, \dots}$ is also an arithmetic sequence with first term $a_{1} + k$ and common difference $d = a_{2} - a_{1} .$

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