UPSKILL MATH PLUS
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Learn moreA sequence is an ordered list of numbers that may follow a particular rule. In this section, we explore a special kind of sequence called an arithmetic progression.
Arithmetic Progression
An arithmetic progression (A.P) is a sequence of numbers in which each term after the first is obtained by adding a fixed number (called the common difference) to the preceding term. Equivalently, the difference between any two consecutive terms is constant
we use \(t_1,\, t_2, \, t_3,\, \ldots\)
where \(t_1\) is a first term, \(t_2\) is a second term, \(t_3\) is a third term.
Common difference
Common difference \(( d )\) \(=\)
This fixed number is called the common difference of the AP. It can be positive, negative or zero.
Example:
1. Consider a sequence \(-5, -1, 3, 7,...\)
\(t_2 - t_1 = -1 - (-5) = -1 + 5 = 4\)
\(t_3 - t_2 = 3 - (-1) = 3 + 1 = 4\)
\(t_4 - t_3 = 7 - 3 = 4\)
\(d = t_2 - t_1 = t_3 - t_2 = t_4 - t_3 \).
Here the common difference is \(4\). Hence the given sequence is an arithmetic progression(A.P).
Arithmetric progression Sequence
Let \(a\) and \(d\) be real numbers. Then the numbers of the form ,... is said to form Arithmetic progression. And it is denoted by \(A\).\(P\), where '\(a\)' is the first term and \(d\) is a common difference.
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 \(t_n = a + (n - 1) d\).
(i) \(n^{\text{th}}\) term using the explicit rule of an A.P.
\(t_n=a+(n-1)d\)
(ii) \(n^{\text{th}}\) term using the recursive rule of an A.P.
\(t_1=a,\ t_n=t_{n-1}+d,\qquad n\ge2\)
(ii) \(n^{\text{th}}\) term using the recursive rule of an A.P.
\(t_1=a,\ t_n=t_{n-1}+d,\qquad n\ge2\)
Important!
To check whether a given term \(t_n\)belongs to an arithmetic progression, substitute \(t_n\) into \(n^t\)\(^h\) term formula and solve for \(n\).
(i) If \(n\) is a natural number \((1,2,3,\ldots)\) then the term lies in the A.P.
(ii) If \(n\) is not a natural number (i.e., a fraction or a decimal), then the term does not lie in the A.P.
Because the position (term number) in a sequence must always be a natural number
Triangular number
The \(n^{\text{th}}\) triangular number is equal to the sum of the first \(n\) natural numbers.
\(T_n=\dfrac{n(n+1)}{2}\)
\(T_n=\dfrac{n(n+1)}{2}\)
Example:
1. Find the \(15^{\text{th}}\) triangular number.
Using the formula,
\(T_{15}=\dfrac{15(15+1)}{2}\)
\(=\dfrac{15\times16}{2}\)
\(=15\times8\)
\(=120\)
Hence, the \(15^{\text{th}}\) triangular number is \(120\).
Using the formula,
\(T_{15}=\dfrac{15(15+1)}{2}\)
\(=\dfrac{15\times16}{2}\)
\(=15\times8\)
\(=120\)
Hence, the \(15^{\text{th}}\) triangular number is \(120\).
2. In a theatre, the first row contains \(12\) seats. Each successive row contains \(2\) more seats than the previous row. Find general \(n^{\text{th}}\) term
Given: First term \((a) = 12\) , Common difference \((d) = 2\)
In general, an arithmetic progression (AP) can be described as \(,...\)
Therefore an arithmetic progression (AP) is \(12, 12 + 2, 12 +2(2), 12 + 3(2), 12 + 4(2),...\)
\(12, 14, 16, 18, 20,...\)
The general \(n^{\text{th}}\) term \(t_n\) of an A.P can be written as \(t_n = a + (n - 1) d\).
\(t_n = 12 + (n - 1) 2\)
\(t_n = 12 + 2n - 2\)
\(t_n = 10 + 2n \)
3. Find the \(n^t\)\(^h\) term of the A.P : \(1.5, 3.5, 5.5, 7.5, …\)
Given: First term: \(a = 1.5,\) Common difference: \( d = 3.5 − 1.5 = 2 \)
Using the Explicit Rule \(t_n = a+(n-1)d\)
\(t_n = 1.5 + (n-1)(2)\)
\(= 1.5 + 2n - 2 \)
\(t_n = 2n - 0.5\)
Using the Recursive Rule \(t_1=1.5, t_n=t_{n-1}+2,\qquad n\ge2.\)
Visualing an AP
Let the number of books arranged on shelves be
\(8,\;14,\;20,\;26,\ldots\), which is an A.P.
Find the \(n^{\text{th}}\) term of the A.P.
\(8,\;14,\;20,\;26,\ldots\), which is an A.P.
Find the \(n^{\text{th}}\) term of the A.P.
\(8,\;14,\;20,\;26,\ldots\), using the explicit rule.
Given: First term: \(a=8\), Common difference: \(d=14-8=6\)
Using the Explicit Rule
\(t_n=a+(n-1)d\)
\(t_n=8+(n-1)(6)\)
\(=8+6n-6\)
\(t_n=6n+2\)
Given: First term: \(a=8\), Common difference: \(d=14-8=6\)
Using the Explicit Rule
\(t_n=a+(n-1)d\)
\(t_n=8+(n-1)(6)\)
\(=8+6n-6\)
\(t_n=6n+2\)
| Shelf Number | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(...\) | \(n\) |
| \(6n+2\) |
\(t_1=6(1)+2\)
\(=6+2\)
\(=8\)
|
\(t_2=6(2)+2\)
\(=12+2\)
\(=14\)
|
\(t_3=6(3)+2\)
\(=18+2\)
\(=20\)
|
\(t_4=6(4)+2\)
\(=24+2\)
\(=26\)
|
\(t_5=6(5)+2\)
\(=30+2\)
\(=32\)
|
\(...\) |
\(t_n=6(n)+2\)
\(=6n+2\)
|
| Number of books | \(8\) | \(14\) | \(20\) | \(26\) | \(32\) | \(...\) | \(6n+2\) |
When we plot the ordered pairs emerging from the table and join the coordinates \((1, 8), (2, 14), (3, 20), (4, 26), (5, 32)\). We observe that they lie on a straight line!
Therefore, the growing pattern of squares represented by a linear pattern
Important!
Every arithmetic progression \((A.P)\) forms a linear pattern and the graphical representation is always a straight line.
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