ARITHMETIC PROGRESSION

An arithmetic progression is a sequence in which the difference between consecutive terms is the same throughout the sequence.

The said difference is called a common difference and this common difference may be positive or negative. The letter d is usually used to denote common difference while $a$ is used to denote the first term of an A.P.

Example 1
$2,6,10,14$.....  are terms of an AP. Find the first term and the common difference.

Solution:
The first term is already given as $2$.

The common difference is easily obtained by subtracting consecutive terms of an A.P.

Thus, $d=12-8=4$

Therefore, the first and common difference is $2$ and $4$ respectively 

The nth term of an A.P
Generally, the nth term of an A.P is usually given by the relation:
$T_n=a+(n-1)d$

With this fact, let's solve the following examples

Example 2
Determine the 7th term of the A.P: 4, 7, 10, 13.....

Solution:
The first is 4, the common difference is $10-7=3$

Using the nth term of an A.P
$T_7=4+(7-1)3$
$T_7=4+18=22$

Therefore, the 7th term of the A.P is $22$

Example 3
What is the 8th term of the sequence 7.1, 6.6, 6.1......

Solution:
The first term is $7.1$, the common difference is $6.6-7.1=-0.5$

$T_8=7.1+(8-1)(-0.5)$
$T_8=7.1+7(-0.5)$
$T_8=7.1-3.5$
$T_8=3.6$

Example 4
Given that 5, x, y, z, 29 are all terms of an A.P. find (x+y)-z.

Solution:
$a=5$
$T_5=29$

If $T_5=a+3d$, then
$a+4d=29$

Also, $a=5$, then
$5+4d=29$
$4d=29-5$
$4d=24$
$\frac{4d}{4}=\frac{24}{4}$
$d=6$

$x=a+d=5+6=11$
$y=a+2d=5+12=17$
$z=a+3d=5+18=23$

$(x+y)-z=(11+17)-23=5$

The sum of the nth term of an A.P
Generally, the sum of the nth term of an A.P is usually given as
$S_n=\frac{n}{2}(2a+(n-1)d)$

Example 5
The first term and common differences of an AP are 67 and 8 respectively. Find the sum of the first five terms of the A.P.

Solution:
$S_4=\frac{5}{2}(2(67)+(4)8)$
$S_4=\frac{5}{2}(134+32)$
$S_4=\frac{5}{2}(166)$
$S_4=\frac{830}{2}$
$S_4=415$

Example 6
The first term of an A.P is 3 and its common difference is 6. Find the sum of the first four terms of the A.P.

Solution:
$S_4=\frac{4}{2}(2(3)+3(6))$
$S_4=2(6+18)$
$S_4=2(24)$
$S_4=48$.

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Example 7
The 8th term of an A.P is 46 while the sum of the first 8 terms is 200. Find the:
1)the  first term
2) common difference
3) sum of the first twelve terms

Solution:
From the first statement
$46=a+(8-1)d$
$46=a+7d$........eqn 1

From the second statement
$200=\frac{8}{2}(2a+(8-1)d)$
$200=4(2a+7d)$
$200=8a+28d$........eqn 2

Solving eqn 1 and 2 simultaneously
$46=a+7d$
$200=8a+28d$

Multiply eqn 1 by 4 and eqn 2 by 1
$4(46=a+7d)$
$1(200=8a+28d)$

Substracting both eqn
$\begin{matrix} 184=4a+28d \\ -(200=8a+28d)\\ \hline 0-16=-4a \end{matrix}$
$\frac{-4a}{-4}=\frac{-16}{-4}$
$a=4$

Substituting 4 for an in eqn 1
$46=4+7d$
$46-4=7d$
$42=7d$
$\frac{42}{7}=\frac{7d}{7}$
$d=6$

Now, let's solve for the sum of the first 12 terms of the A.P.
$S_{12}=\frac{12}{2}(2(4)+(12-1)6)$
$S_{12}=6(8+66)$
$S_{12}=6(74)$
$S_{12}=444$

Therefore, 
1. The first term of the A.P is 4
2. The common difference is 6
3. The sum of the first twelve terms of the A.P is 444.
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