Terms and common difference of an A.P. — lesson. Mathematics State Board, Class 10.

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In the previous theory, we understood that Arithmetic Progress $A.P$ formed a sequence of $a$, $a + d$, $a + 2d$, $a + 3d$, $a + 4d$, $a + 5d$. Here each number is called a term.

The first term is '$a$', the second term is '$a + d$' which is obtained by adding the difference $(d)$, and the third term is '$a+2d$'.

The terms of an $A.P.$ can be written several ways. Now let's see the few ways which are:

General $n^t$$^h$ term:

When $n ∈ N$, $ n = 1, 2, 3, 4, ...$

$t_1 = a = a + (1 - 1) d$

$t_2 = a + d = a + (2 - 1) d$

$t_3 = a + 2d = a + (3 - 1) d$

$t_4 = a + 3d = a + (4 - 1) d$

Here '$t$' refers to terms, and '$n$' denotes the number of terms.

In general, the $n^t$$^h$ term denoted by $t_n$ can be written as $t_n = a + (n - 1) d$.

In a finite $A.P.$ whose first term is '$a$' and last term '$l$', then the number of terms in the $A.P.$ is given by

l = a + (n - 1) d \Rightarrow n = (\frac{l - a}{d}) + 1

Common difference:

To find the common difference of an $A.P$., we should generally subtract the first term from the second term, the second from the third and so on.

The first term $t_1 = a$ and the second term $t_2 = a + d$.

Difference between $t_1$ and $t_2$ is $t_2 - t_1 = (a + d) - a = d$.

Similarly, $t_2 = a + d$ and $t_3 = a + 2d$.

Therefore, $t_3 - t_2 = a + 2d - a + d = d$.

So, in general $d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4$.

Thus,

d = t_{n} - t_{n - 1}

where $ n = 1, 2, 3, …$

The common difference of an $A.P.$ can be positive, negative, or zero.

Example:

1. Consider an $A.P.$ $10, 13, 16, 19, 22, ...$

$d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4$.

$d = 13-10 = 16-13 = 19-16 = 22-19 =3$.

Here the common difference is $3$.

2. Take an $A.P.$ $-7, -10, -13, -16, ...$

$d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4$.

$d = -10-(-7) = -13-(-10) = -16-(-13) = -3$.

Here the common difference is $-3$.

3. If an $A.P$ is $-7, -7, -7, -7, -7, ...$

$d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 = t_5 - t_4$.

$d = -7-(-7) = -7-(-7) = -7-(-7) = -7-(-7) =0$.

Here the common difference is $0$.

An arithmetic progression having a common difference of zero is called a constant arithmetic progression. For example, here the $A.P$ $-7, -7, -7, -7, -7, ...$ is called constant arithmetic progression.

Condition for three numbers to be in $A$.$P$.

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

So, $a + c$ $= 2a + 2d = 2 (a + d) = 2b$

Thus, $2b = a + c$

Similarly, if $2b = a +c$, then $b − a = c −b$ so $a, b, c$ are in $A.P.$

Thus three non-zero numbers $a, b, c$ are in $A$.$P$. if and only if $2b = a + c$

Example:

If 3$+$ $x$, 18$-$ $x$, 5$x$ $+$ 1 are in $A$.$P$. then find $x$.

Since the given $A$.$P$. series has three numbers, we can use the above-derived expression $2b = a + c$.

Let us take

\begin{array}{l} \underset{⏟}{3 + x}, \underset{⏟}{18 - x}, \underset{⏟}{5 x + 1} \\ a b c \end{array}

Now substitute the known values in the expression $2b = a + c$.

\begin{array}{l} 2 (18 - x) = 3 + x + 5 x + 1 \\ 36 - 2 x = 4 + 6 x \\ 36 - 4 = 6 x + 2 x \\ 32 = 8 x \\ \frac{32}{8} = x \\ x = 4 \end{array}

The value of $x$ $=$ 4

Important!

Key takeaways:

The common difference of an $A.P.$ can be positive, negative, or zero.
The common difference of constant $A.P.$ is zero.
If '$a$' and '$l$' are the first and last terms of an $A.P.$ then the number of terms $(n)$ is $n = (\frac{l - 1}{d}) + 1$

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2. Terms and common difference of an A.P.

Theory: