Question

# Write the arithmetic progression when the first term $a$ and common difference $d$ are $a = 4,{\text{ }}d = - 3$.

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Hint:In the arithmetic progression, for a given series, the terms used are the first term, the common difference between the two terms and the nth term. Suppose, ${a_1},{a_2},{a_{3,}}...,{a_n}$is an arithmetic progression then the common difference $d$can be obtained as:
$d = {a_2} - {a_1} = {a_3} - {a_2} = ... = {a_n} - {a_{n - 1}}$
Substitute the values of the first term and common difference we get in the series.

It is given that: The first term $a$ and common difference $d$ of the arithmetic progression are $a = 4,{\text{ }}d = - 3$ .
We have to write the arithmetic progression.
A progression is a special is a special type of sequence for which it is possible to a formula for the nth term. The arithmetic progression is the most commonly used sequence in the world of Mathematics.
An Arithmetic progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
In the arithmetic progression, for given series, the term used are the first term, the common difference between the two terms and the nth term. Suppose, ${a_1},{\text{ }}{a_2},{\text{ }}{a_{3,}}{\text{ }}...,{\text{ }}{a_n}$ is an arithmetic progression then the common difference $d$ can be obtained as:
$d = {a_2} - {a_1} = {a_3} - {a_2} = ... = {a_n} - {a_{n - 1}}$
The arithmetic progression can be written as: $a,{\text{ }}a + d,{\text{ }}a + 2d,{\text{ }}a + 3d,{\text{ }}...,{\text{ }}a + (n - 1)d$.
Substitute the values $a = 4,\;d = - 3$ in the given series we get,
$4,{\text{ }}[4 + ( - 3)],{\text{ }}[4 + 2( - 3)],{\text{ }}[4 + 3( - 3)],{\text{ }}[4 + 4( - 3)],{\text{ }}...$
Simplifying we get,
The given series is $4,{\text{ }}1,\, - 2,{\text{ }} - 5,{\text{ }} - 8,{\text{ }}...$
Hence, the arithmetic progression is $4,{\text{ }}1,\, - 2,{\text{ }} - 5,{\text{ }} - 8,{\text{ }}...$

Note:The fixed number that must be added to any term of an arithmetic progression to get the next term is known as the common difference of the series. In a geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio.