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Arithmetic Progressions
1.0Sequence
In our daily life, we come across the arrangement of numbers or objects in an order such as arrangement of students in a row as per their roll numbers, arrangement of books in the library, etc. An arrangement of numbers depends on the given rule
Thus, a sequence is an ordered arrangement of numbers according to a given rule. Terms of a sequence : The individual numbers that form a sequence are the terms of a sequence. For example : 2, 4, 6, 8, 10,.... forming a sequence are called the first, second, third, fourth, fifth,.... terms of the sequence. The terms of a sequence in successive order is denoted by ' ' or ' '. The term ' ' is called the general term of the sequence. Series The sum of terms of a sequence is called the series of the corresponding sequence. . is an infinite series, where as is a finite series of terms. Usually, the series of finite number of terms is denoted by .
Numerical Ability 1 Write the first five terms of the sequence, whose term is . Solution: Substituting and 5 , we get ; ; Thus, the required terms are : and 0 .
2.0Progression
It is not always possible to write each and every sequence of some rule.
3.0Arithmetic progressions
- We know about some sequences which are arithmetic progressions as.
(i) Sequence of even numbers:
,
(ii) Sequence of odd numbers: , ...
(iii) Sequence of multiples of 5: ,
(iv) Sequence of number which are divisible by 7 and give remainders 1 in each case:
,
An arithmetic progression is list of numbers (terms) in which the first term is given and each term, other than the first term is obtained by adding a fixed number 'd' to the preceding term.
The fixed number ' d ' is known as the common difference of the arithmetic progression. Its value can be positive, negative or zero.
The first term is denoted by 'a' or 'a1' and the last term by ' '. e.g., Consider a sequence , .....
Here,
A sequence of non-zero numbers is said to be a geometric sequence or GP, if i.e. if constant for all n. e.g., 3 , SPOT LIGHT
Therefore, the sequence is an arithmetic progression in which the first term and the common difference .
Symbolic form : Let us denote the first term of an AP by , second term by term by and the common difference by . Then the AP becomes .
So, . General form : In general form, an arithmetic progression with first term 'a' and common difference ' ' can be represented as follows: .
Finite AP : An AP in which there are only a finite number of terms is called a finite AP. It may be noted that each such AP has a last term.
e.g., (a) The heights (in cm) of some students of a school standing in a queue in the morning assembly are . (b) The minimum temperatures (in degree Celsius) recorded for a week in the month of January in a city arranged in ascending order are
Infinite AP : An AP in which the number of terms is not finite is called infinite AP. That means infinite AP does not have a last term.
The sum of cubes of first n natural numbers i.e., is usually written as .
e.g., (a) (b)
Least information required : To know about an AP, the minimum information we need to know is the first term a and the common difference d. For instance if the first term a is 6 and the common difference is 3 , then AP is . Similarly, when , the AP is , the AP is So, if we know what a and d are, we can list the AP.
- If a constant is added or subtracted from each term of an AP then the resulting sequence is also an AP with the same common difference.
In which of the following situations, the list of numbers obtained will be in the form of an arithmetic progression? (i) Number of students left in the school auditorium from the total strength of students when they leave the auditorium in batches of 25 . (ii) The amount of air present in the cylinder when a vacuum pump removes each time of the air left in the cylinder. (iii) Cash price of a particular brand of washing machine in the market is . Sawitri buys one washing machine on monthly instalments of plus an interest at the rate of per month on the balance amount. She makes the first instalment after one month. No amount is to be paid at the time of purchase. Explanation: (i) and so on. Thus, the list of numbers is as below : 1000, 975, 950, 925, ...... Here, . Therefore, the above list of numbers forms an AP (ii) Let us suppose that air present in the cylinder in the beginning is x units. Every time the vacuum pump removes the of the air present in the cylinder. Here, units, units units, units units units units, units units units and so on.
In a finite AP the sum of the terms equidistant from the beginning and end is always same and is equal to the sum of first and last term.
Thus the list of numbers is as below : Here, and so on. i.e., . Obviously, the common differences of two consecutive terms are not same throughout. Hence, the list of numbers does not form an AP. (iii) , , , and so on. Thus, the list of the amounts of instalments is (in rupees) 1240, 1220, 1200, 1180, .... and so on. Hence, the list form an AP.
Numerical Ability 2 Write first four terms of an AP, when the first term a and the common difference are as follows : . Solution: First term, Second term Third term Fourth term Hence, first four terms of the given AP are 4, 9, 14, 19. General term of an arithmetic progression The formula for writing general term or the term of
r^{\text {th }} term of a finite arithmetic progression from the end Let there be an arithmetic progression with first term a and common difference d. If there are terms in the arithmetic progression, then term from the end
- If is the last term of an arithmetic progression then term from the end is the term of an arithmetic progression whose first term is and common difference is -d . term from the end
- Whenever we find number of terms(n) for any arithmetic progression, then n should be natural number only.
Numerical Ability 3 Find the term of the sequence : 7, 3, . Solution The given sequence is an AP in which the first term and the common difference . Hence, .
Numerical Ability 4 Find the term from the end of the AP 17, 14, 11, ...., . Solution: The given AP : 17, 14, 11, ..., -40 Here, Let there be terms in the given AP. Then, term Hence, there are 20 terms in the given AP. Now, 6th term from the end term from the end Hence, the term from the end of the given AP is -25 . Alternate solution term from end
A sequence is an AP if the sum of its first terms is of the form , where A, B are constants independent of . In such a case, the common difference is 2A.
4.0Selection of terms in an AP
Sometimes we require certain number of terms in AP whose sum is given. The following ways of selecting terms are generally very convenient.
It should be noted that in case of an odd number of terms, the middle term is a and the common difference is while in case of an even number of terms the middle terms are and the common difference is 2 d .
- Selection of terms is useful when sum of certain terms are given. If the sum of terms is not given, then select terms as .
- If three numbers in order are in AP. Then Common difference Thus, are in AP if and only if .
- If are in AP, then is known as the arithmetic mean (AM) between and .
- If are in AP then, . Thus, AM between a and b is .
Numerical Ability 5 The sum of three numbers in AP is , and their product is 8 . Find the numbers. Solution: Let the three numbers in AP be . Then, Sum Now, product If , the numbers are . If , the numbers are . Thus, the numbers are or .
Numerical Ability 6 If are in AP, find the value of . Solution: As, are in AP,
5.0Sum of first \mathbf{n} terms of an arithmetic progression
The sum ' ' of n terms of an arithmetic progression with first term 'a' and common difference ' d ' is or Where last term.
- In the formula , there are four quantities viz. and d. If any three of these are known, the fourth can be determined. Sometimes, two of these quantities are given. In such a case, remaining two quantities are provided by some other relation.
- If the sum of terms of a sequence is given, then term of the sequence can be determined by using the following formula i.e., the term of an AP is the difference of the sum of first terms and the sum of first ( ) terms of it.
Numerical Ability 7 Find the sum of the first 20 terms of the AP : 5, 8, 11, 14, ..... . Solution First term of the , i.e., Common difference , i.e., and
Numerical Ability 8 Find the sum : Solution: As the given AP is Here, Let the number of terms of the AP be . We know that Hence, the required sum is 286.
Numerical Ability 9 Find the number of terms of the AP 54, 51, 48,...so that their sum is 513. Solution: The given AP is Here, Let the sum of terms of this AP be . We know that (Dividing throughout by 3 ) or Hence, the sum of 18 terms or 19 terms of the given AP is 513. So, or 19 Note: Here 19th term
Numerical Ability 10 Rakesh has to buy a TV. He can buy TV either making cash down payment of ₹ at once or by making 14 monthly instalments as below : ₹ 1500 ( month), ₹ 1450 ( month), ₹ 1400 ( 3 rd month), ₹ 1350 ( 4 th month), ..... Each instalment except the first is ₹ less than the previous one. Find: (i) Amount of the instalment paid in the month. (ii) Total amount paid in 14 instalments. (iii) How much extra he has to pay in addition to the amount of cash down payment? Solution: . Now, . So, the fourteen monthly instalments form an AP. Here, and (i) Thus, instalment (ii) Here, total amount paid = ₹ 16,450 (iii) Extra paid amount (i.e., in addition to cash down payment)