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:**

$S_4=\frac{4}{2}(2(3)+3(6))$

$S_4=2(6+18)$

$S_4=2(24)$

$S_4=48$.

**Related posts****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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