On this page you will find few practice question in Mathematics (Arithmetic Progression) which will enhance your preparation for the forthcoming GCE Exam.
TOPIC: ARITHMETIC PROGRESSION
Watch the video tutorial below, for detailed explanation of Arithmetic Progression, then answer the questions that follows, below the video.
………..
Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. It is also called Arithmetic Sequence.
For example: The sequence 3, 5, 7, 9, 11,… is an arithmetic progression
with common difference 2.
The difference between any two successive members is called common difference.
For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, … has
- a = 1 (the first term)
- d = 4 (the “common difference” between terms)
In general an arithmetic sequence can be written like: {a, a+d, a+2d, a+3d, … }.
Arithmetic progression is defined as the sequence of numbers in algebra such that the difference between every consecutive term is the same. It can be obtained by adding a fixed number to each previous term.
AP Formulae
The nth Term of AP Formula
The formula for finding the nth term of an AP is:
Here,
a = First term
d = Common difference
n = Number of terms
an = nth term
Example: Find the nth term of AP:
5, 8, 11, 14, 17, …, an, if the number of terms are 12.
Solution:
AP: 5, 8, 11, 14, 17, …, an (Given)
n = 12
By the formula we know, an = a + (n – 1)d
First-term, a = 5
Common difference, d = (8 – 5)= 3
Therefore, an = 5 + (12 – 1)3
= 5 + 33
= 38
Sum of n Terms of AP Formula
For an AP, the sum of the first n terms can be calculated if the first term and the total number of terms are known. The formula for the sum of AP is:
Here,
S = Sum of n terms of AP
n = Total number of terms
a = First term
d = Common difference
Arithmetic Progression Sum Formula When First and Last Terms are Given:
When we know the first and last term of an AP, we can calculate the sum of the arithmetic progressions using this formula:
Sn = n/2(a + l)
Derivation:
Consider an AP consisting “n” terms having the sequence a, a + d, a + 2d, … , a + (n – 1) × d
Sum of first n terms = a + (a + d) + (a + 2d) + ………. + [a + (n – 1) × d] —— (i)
Writing the terms in reverse order, we get:
S = [a + (n – 1) × d] + [a + (n – 2) × d] + [a + (n – 3) × d] + ……. (a) —— (ii)
Adding both the equations term wise, we have:
2S = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + … + [2a + (n – 1) ×d] (n-terms)
2S = n × [2a + (n – 1) × d]
S = n/2[2a + (n − 1) × d]
Let’s understand this formula with examples:
Example 1: Find the sum of the following arithmetic progression:
9, 15, 21, 27, … The total number of terms is 14.
Solution:
AP = 9, 15, 21, 27, …
We have: a = 9,
d = (15 – 9) = 6,
and n = 14
By the AP sum formula, we know:
S = n/2[2a + (n − 1) × d]
= 14/2[2 x 9 + (14 – 1) x 6]
= 14/2[18 + 78]
= 14/2 [96]
= 7 x 96
= 672
Hence, the sum of the AP is 672.
Example 2: Find the sum of the following AP: 15, 19, 23, 27, … , 75.
Solution: AP: 15, 19, 23, 27, … , 75
We have: a = 15,
d = (19 – 15) = 4,
and l = 75
We have to find n. So, using the formula: l = a + (n – 1)d, we get
75 = 15 + (n – 1) x 4
60 = (n – 1) x 4
n – 1 = 15
n = 16
Here the first and last terms are given, so by the AP sum formula, we know:
S = n/2[first term + last term]
Substituting the values, we get:
S = 16/2 [15 + 75]
= 8 x 90
= 720
Hence, the sum of the AP is 720.
AP: MATHS WAEC GCE PRACTICE QUESTIONS
NOTE: TYPE YOUR ANSWERS INTO THE COMMENT BOX BELOW THESE QUESTIONS, THE CORRECTIONS WILL BE POSTED SOON.
1. The nth term of a sequence is Tn = 5 + (n – 1)2. Evaluate T4 – T6
- A. 30
- B. 16
- C. -16
- D. -30
2. The nth term of a sequence is n2 – 6n – 4. Find the sum of the 3rd and 4th terms.
- A. 24
- B. 23
- C. -24
- D. -25
3. Find the 19th term of the A.P. 5656, 8686, 116116……………..
- A. 71/2
- B. 9
- C. 9 1/2
- D. 9 5/6
- E. 10
4. If the 3rd and the 5th terms of an A.P are 6 and 10 respectively, find the 1st term and the common difference respectively.
- A. 1, 2
- B. 2, 2
- C. 2, 3
- D. 3, 2
- E. 3,3
5. Find the 21st term of the Arithmetic Progression (A.P.): -4, -1.5, 1, 3.5,…
- A. 43.5
- B. 46
- C. 48.5
- D. 51
