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Arithmetic Progression - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Arithmetic Progression, Important Properties of an AP -Part 1, Important Properties of an AP -Part 2 is considered one of the most asked concept.
44 Questions around this concept.

Solve by difficulty

The $20^{\text {th }}$ term from the end of the progression $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}$ is:

-118

-110

-115

-100

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Let $f\left ( x \right )$ be a polynomial function of second degree. If $f\left (1 \right )= f\left (-1 \right )$ and $a,b,c$ are in A.P., then ${f}'\left ( a \right ),{f}'\left ( b \right ),{f}'\left ( c \right )$ are in :

G.P.

H.P.

Arithmetic Geometric Progression

A.P.

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Concepts Covered - 3

Arithmetic Progression

Arithmetic Progression

An arithmetic progression is a sequence in which each term increases or decreases by a constant term or fixed number. This fixed number is called the common difference of an AP and generally denoted by ‘d’.

$\\\mathrm{If\;\mathit{a_1,a_2,a_3,a_4........a_{n-1},a_n}\;are\;in\;AP}\\\mathrm{Then}\\\mathrm{\mathit{d=a_2-a_1=a_3-a_2=............=a_n-a_{n-1}}}$

Eg, 1, 4 , 7, 10,.... is an AP with common difference 3

Also, 2, 1, 0, -1,.... is an AP with common difference -1

In AP, first term is generally denoted by ‘a’

General Term of an AP

We found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence. First, write the first few terms of a sequence where the first term is ‘a’ and the common difference is ‘d’. We will then look for a pattern.

i.e. a, a + d, a + 2d, a + 3d, ………

Then the n^th term (general term) of the A.P. is $\mathrm{\mathit{a_n=a+(n-1)d}}$ .

$\\\mathrm{\mathit{a_1=a+(1-1)d=a}}\\\mathrm{\mathit{a_2=a+(2-1)d=a+d}}\\\mathrm{\mathit{a_3=a+(3-1)d=a+2d}}\\\mathrm{\mathit{a_4=a+(4-1)d=a+3d}}\\\mathrm{...}\\\mathrm{...}\\\mathrm{\mathit{a_n=a+(n-1)d=l=last\;term}}$

On simplification of general term, we can see that the general term of an AP is always a linear in n

$T_n=an+b$

Important Properties of an AP -Part 1

Important Properties of an AP

If a fixed number is added to or subtracted from each term of a given A.P., then the resulting series is also an A.P. and its common difference remains the same.
If each term of an A.P. is multiplied by a fixed constant or divided by a non-zero fixed constant then the resulting series is also in A.P.
If $a_1,a_2,a_3....$ and $\mathit{b_1,b_2,b_3....}$ are two A.P’s, then ${a_1\pm b_1,a_2\pm b_2,a_3\pm b_3....}$ are also in A.P.

Important Properties of an AP -Part 2

Important Properties of an AP

4. If terms of an A.P. are taken at equal intervals, then the new sequence formed as also an A.P.

Eg, If from A.P., 1, 3, 5, 7, 9, 11, 13,.... we take terms at equal intervals, lets say first, third, fifth, seventh,....terms, then the resultant sequence will be

1, 5, 9, 13, ....which is also an A.P.

$\\\mathrm{If\;\mathit{a_1,a_2,a_3,....,a_n}\;are\;in\;A.P.,\;then}\\\\\mathrm{\mathit{a_r=\frac{a_{r-k}+a_{r+k}}{2},\forall \;k,\;0\leq k\leq n-r}}$

6. Choosing terms in A.P.

If sum of few terms (like 3, 4, or 5 terms) in an A.P. is given in the problem, then selecting the following terms reduces the calculation

If we need to choose three terms in an A.P., then choose (a-d), a, (a+d) [Note: Here first term is a-d, and common difference is d]
If we need to choose four terms in an A.P., then choose (a-3d), (a-d), (a+d), (a+3d) [Note: Here first term is a-3d, and common difference is 2d]
If we need to choose five terms in an A.P., then choose (a-2d), (a-d), a, (a+d), (a+2d) [Note: Here first term is a-2d, and common difference is d]

7. Sum of terms equidistant from beginning and end of an AP is constant and it equals sum of first and the last terms.

$\mathit{a_1+a_n=a_2+a_{n-1}=a_3+a_{n- 2}=......=a_r+a_{n-r+1}}$