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.

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**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, …, a_{n}, if the number of terms are 12.

**Solution:**

AP: 5, 8, 11, 14, 17, …, a_{n} (Given)

n = 12

By the formula we know, **a**_{n}** = a + (n – 1)d**

First-term, a = 5

Common difference, d = (8 – 5)= 3

Therefore, a_{n} = 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 n^{th} term of a sequence is T_{n} = 5 + (n – 1)^{2}. Evaluate T_{4} – T_{6}

**A.**30**B.**16**C.**-16**D.**-30

2. The nth term of a sequence is n^{2} – 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