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Example 9
Start with a number, say -3, and keep on adding a different number, say
.Solution
AP:
Example 10
Start with the function, say cos x, and keep on adding the number, say 2.Solution
AP: cos x, cos x + 2, cos x + 4, cos x + 6,...
[B] The General AP
Start with a and keep on adding the number d to generate the AP: a, a + d, a + 2d, a + 3d, a + 4d, ....
Notes:
(i) The different terms:
u1 = a = the first term,
u2 = a + d = the second term,
u3 = a + 2d = the second term,
u45 = a + 44d
(ii) The General Term: un = a + (n - 1)d
(iii) The difference d between any term and the previous one is a constant known as the common difference. This is the test for an AP.
Test: A sequence is an AP iff un+1 – un = constant = d for all
(iv) The sum Sn of an AP
The formula for adding up the first n terms of an AP is given by:
(v) If you know the first term u1 = a of an AP and the common difference d you can find everything else.
(vi) Make sure you know what all the symbols mean.
u1 = a = the value of the first term
d = common difference = any term - previous term
un = the value of the object in the nth place
n = the place in the list
Sn = the sum of the first n terms
[C] Tricks for 3 consecutive terms in an AP.
(i) If you are asked to choose 3 consecutive terms in an AP then choose them as: a - d, a, a + d
(ii) If you are told that a, c, b are consecutively in an AP then simply write:
(iii) The Arithmetic Mean (AM) i.e. the ordinary average
Given 2 numbers a, b can you find a number between these so that all three are consecutively in an AP.
From (ii) the AM =
Example 11
The second term of an AP is 4 and the sixth term is 16. Find the first 4 terms.Solution
Example 12
Which term is 84 in the AP: 12, 15, 18, 21,....Solution
Example 13
If 5, x, y, 32 are four consecutive numbers of an AP find x and y.Solution
Example 14
How many terms of the AP: 12, 16, 20,... must be added to give 208.Solution
Example 15
Find the sum of all even numbers between 103 and 435 which are divisible by 3.Solution
Example 16
Find three consecutive numbers in an AP whose sum is 6 and whose product is -24.Solution
Example 17
If
are consecutive terms of an AP show that 
are also consecutive terms of an AP.Solution
