Arithmetic Progression - Practice Questions & MCQ

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 is generally denoted by 'd'.

If $a_1, a_2, a_3, a_4 \ldots \ldots . a_{n-1}, a_n$ are in AP
Then

$
d=a_2-a_1=a_3-a_2=\ldots \ldots \ldots \ldots=a_n-a_{n-1}
$

$\mathrm{Eg}, 1,4,7,10, \ldots$ is an AP with common difference 3
Also, $2,1,0,-1, \ldots$ is an AP with common difference -1
In AP, the 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. $a, a+d, a+2 d, a+3 d$, $\qquad$
Then the $\mathrm{n}^{\text {th }}$ term (general term) of the A.P. is $a_n=a+(n-1) d$.

$
\begin{aligned}
& a_1=a+(1-1) d=a \\
& a_2=a+(2-1) d=a+d \\
& a_3=a+(3-1) d=a+2 d \\
& a_4=a+(4-1) d=a+3 d \\
& \cdots \\
& \cdots \\
& a_n=a+(n-1) d=l=\text { last term }
\end{aligned}
$
On simplification of a general term, we can see that the general term of an AP is always linear in $n$

$
T_n=a n+b
$
 

Important Properties of an AP
1. 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.
2. 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.
3. If $a_1, a_2, a_3 \cdots$ and $b_1, b_2, b_3 \cdots$ are two A.P's, then $a_1 \pm b_1, a_2 \pm b_2, a_3 \pm b_3 \cdots$ are also in A.P.

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

Eg, If from A.P., 1,3,5,7,9,11,13,...., we take terms at equal intervals, let's say first, third, fifth, seventh, ...terms, then the resultant sequence will be $1,5,9,13, \ldots$ is also an A.P.

If $a_1, a_2, a_3, \ldots, a_n$ are in A.P., then
5. $\quad a_T=\frac{a_{rk}+a_{r+k}}{2}, \forall k, \quad O \leq k \leq nr$
6. Choosing terms in A.P.

If the sum of a 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 (ad), $a_1(a+d) \quad$ [Note: Here the first term is $ad$, and the common difference is $d$ ]
If we need to choose five terms in an A.P., then choose (a2d), (ad), a, (a+d), (a+2d) [Note: Here the first term is a2d, and the common difference is $d]$
7. The sum of terms equidistant from the beginning and end of an $A P$ is constant and it equals the sum of the first and the last terms.

$
a_1+a_n=a_2+a_{n1}=a_3+a_{n2}=\ldots \ldots=a_r+a_{nr+1}
$

8. If $a, b, c$ are in A.P., then $2 b=a+c$