Question

motivation for studying arithmetic progression?​

Answer 1

Answer:

Arithmetic Progression

Step-by-step-explanation:

Arithmetic Progression or AP is a concept in Mathematics.

It is related to sequences and series.

The basic terms in the AP are first term, common difference, $\displaystyle{\sf\:n^{th}}$ term, number of terms and sum of the terms.

1. Sequence:

1. A set of numbers or mathematical terms in a definite order is called as a sequence.

Example:

1, 2, 3, 4, ...... OR Set of all natural numbers is a sequence.

2, 4, 6, 8, ..... OR Set of all even natural numbers is a sequence.

2. Every term in the sequence has a specific place.

Here, 2 is on the $\displaystyle{\sf\:1^{st}}$ place, 4 is on the $\displaystyle{\sf\:2^{nd}}$ place and continuing further 100 is on the $\displaystyle{\sf\:50^{th}}$ place in the given sequence.

3. A sequence can be represented as $\displaystyle{\sf\:a_1\:,\:a_2\:,\:a_3\:,\:\dots\:,\:a_n}$.

2. Common Difference:

1. The difference between two consecutive terms in a sequence is called as Common Difference.

2. It is denoted by letter "d".

3. For a sequence $\displaystyle{\sf\:a_1\:,\:a_2\:,\:a_3\:,\:\dots\:,\:a_n}$, common difference d is -

$\displaystyle{\sf\:d\:=\:a_2\:-\:a_1}$

3. Arithmetic Progression:

1. If the common difference between two consecutive terms of a sequence is constant, then the sequence is called as Arithmetic Progression or AP.

2. In a sequence, $\displaystyle{\sf\:t_1\:,\:t_2\:,\:t_3\:,\:\dots\:,\:t_n}$, if common difference $\displaystyle{\sf\:t_{n\:+\:1}\:-\:t_n}$ is constant, then the sequence is an AP.

3. The common difference in an AP can be any integer or zero.

3. $\displaystyle{\sf\:n^{th}}$ term of an AP:

The place of a particular term in an AP is called as $\displaystyle{\sf\:n^{th}}$ term of an AP.

For an AP 2, 5, 8, 11, 14, .... the place of 14 is 5.

Hence, $\displaystyle{\sf\:5^{th}}$ term of the AP is 14.

4. Formula for $\displaystyle{\sf\:n^{th}}$ term of an AP:

$\displaystyle{\boxed{\red{\sf\:t_n\:=\:a\:+\:(\:n\:-\:1\:)\:d}}}$

5. Derivation of the formula for $\displaystyle{\sf\:n^{th}}$ term of an AP:

Consider an AP $\displaystyle{\sf\:t_1\:,\:t_2\:,\:t_3\:,\:t_4\:,\:\dots\:,\:t_n}$ where first term is "a" and common difference is "d", then

$\displaystyle{\sf\:t_1\:=\:a}$

$\displaystyle{\implies\sf\:t_2\:=\:t_1\:+\:d\:=\:a\:+\:d\:=\:a\:+\:1\:\times\:d\:=\:a\:+\:(\:2\:-\:1\:)\:d}$

$\displaystyle{\implies\sf\:t_3\:=\:t_2\:+\:d\:=\:a\:+\:d\:+\:d\:=\:a\:+\:2d\:=\:a\:+\:2\:\times\:d\:=\:a\:+\:(\:3\:-\:1\:)\:d}$

$\displaystyle{\implies\sf\:t_4\:=\:t_3\:+\:d\:=\:a\:+\:2d\:+\:d\:=\:a\:+\:3d\:=\:a\:+\:3\:\times\:d\:=\:a\:+\:(\:4\:-\:1\:)\:d}$

From this, we get,

$\displaystyle{\pink{\sf\:t_n\:=\:a\:+\:(\:n\:-\:1\:)\:d}}$

6. Sum of first n terms of an AP:

The addition of either all the terms or particular number of terms in an AP is called as sum of "n" terms of an AP.

7. Formula for sum of first n terms of an AP:

$\displaystyle{\boxed{\red{\sf\:S_n\:=\:\dfrac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}}}$

8. Derivation of the formula for sum of first n terms of an AP:

Consider an AP $\displaystyle{\sf\:a\:,\:a\:+\:d\:,\:a\:+\:2d\:,\:+\:a\:+\:3d\:,\:\dots\:,\:t_n}$

It can be written as $\displaystyle{\sf\:a\:,\:a\:+\:d\:,\:a\:+\:2d\:,\:+\:a\:+\:3d\:,\:\dots\:,\:a\:+\:(\:n\:-\:1\:)\:d}$

Here, "a" is the first term and "d" is the common difference.

Let $\displaystyle{\sf\:S_n}$ be the sum of first n terms of the AP.

$\displaystyle{\sf\:S_n\:=\:a\:+\:[\:a\:+\:d\:]\:+\:\dots\:+\:[\:a\:+\:(\:n\:-\:2\:)\:d\:]\:+\:[\:a\:+\:(\:n\:-\:1\:)\:d\:]\:\:-\:-\:-\:(\:1\:)}$

It can be also written as -

$\displaystyle{\sf\:S_n\:=\:[\:a\:+\:(\:n\:-\:1\:)\:d\:]\:+\:[\:a\:+\:(\:n\:-\:2\:)\:d\:]\:+\:\dots\:+\:[\:a\:+\:d\:]\:+\:a\:\:\:\:-\:-\:-\:(\:2\:)}$

Adding equations ( 1 ) & ( 2 ), we get,

$\displaystyle{\sf\:S_n\:+\:S_n\:=\:a\:+\:[\:a\:+\:(\:n\:-\:1\:)\:d\:]\:+\:[\:a\:+\:d\:]\:+\:\dots\:+\:[\:a\:+\:(\:n\:-\:2\:)\:d\:]\:+\:[\:a\:+\:(\:n\:-\:2\:)\:d\:]\:+\:[\:a\:+\:d\:]\:+\:[\:a\:+\:(\:n\:-\:1\:)\:d\:]\:+\:a}$

$\displaystyle{\implies}\sf\:2\:S_n\:=\:[\:a\:+\:a\:+\:(\:n\:-\:1\:)\:d\:]\:+\:[\:a\:+\:a\:+\:(\:n\:-\:2\:)\:d\:]\:+\:\dots\:+\:[\:a\:+\:(\:n\:-\:2\:)\:d\:+\:a\:+\:d\:]\:+\:[\:a\:+\:(\:n\:-\:1\:)\:d\:+\:a\:]$

$\displaystyle{\implies\sf\:2\:S_n\:=\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]\:+\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]\:\dots\:+\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]\:\dots\:\times\:n}$

$\displaystyle{\implies\sf\:2\:S_n\:=\:n\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}$

$\displaystyle{\implies\pink{\sf\:S_n\:=\:\dfrac{n}{2}\:[\:2a\:+\:(\:n\:-\:1\:)\:d\:]}}$

If "a" is first term and "l" is last term of an AP, then the formula for the sum of first n terms of an AP is -

$\displaystyle{\boxed{\red{\sf\:S_n\:=\:\dfrac{n}{2}\:[\:a\:+\:l\:]}}}$

Arithmetic Progression can be studied this way.