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Wednesday, 4 November 2015

Arithmetic Progressions questions

 Arithmetic Progressions  questions

A list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term is called Arithmetic Progressions. This fixed number is called the common difference of the AP .

The common difference is denoted by d and the first term by a.   

Then the AP beco mes:  a, a + d  a + 2d,  a + 3d, . . .,
T1 = a      ; T2 =  a+ d ;  T3 = a +2d  ; T4 = a + 3d ……….Tn = a + (n-1)d

So, the nth term an of the AP with first term a and common difference d is given by an = a + (n – 1) d.

So, T2 – T1 = T3 – T2 = . . . = an – a(n – 1) = d.

an is also called the general term of the AP.

 If there are m terms in the AP, then am represents the last term which is sometimes also denoted by l

Arithmetic Progressions having finite number of terms is called a finite AP.Eg. The heights ( in cm ) of some students of a school standing in a queue in the morning assembly are 147 , 148, 149, . . ., 157.

Arithmetic Progressions  having  (APs) has finite number of terms  is called infinite Arithmetic Progressions. Such APs do not have a last term.
Eg. The minimum temperatures ( in degree celsius ) recorded for a week in the month of January in a city, arranged in ascending order are   – 3.1, – 3.0, – 2.9, – 2.8, – 2.7, – 2.6, – 2.5

Q. 1. Determine k so that k + 2, 4k - 6 and 3k - 2 are three consecutive terms of an AP.

Q. 2. If m th term of A.P. is , and nth term is , show that the mn th terms is 1. 

Q. 3. The first, second and the last terms of an AP are p, q and 2p respectively. Show that its sum is [3pq]/[2(q-p)]. 

Q. 4. A circle is completely divided into n sectors in such a way that the angles of the sectors are in arithmetic progression. If the smallest-of these angles is 8° and the largest 72°, calculate n and the angle in the fourth sector. 

Q. 5. Which term of AP: 3, 10, 17 ... will be 84 more than its 13th term? 

Q. 6. If 9th term of an AP is zero, prove that 29th term is double the 19th term. 

Q. 7. Find a, b such that 27, a, b - 6 are in A.P.

Q. 8. For what value of n, the nth terms of the sequences 3, 10, 17,... and 63, 65, 67,... are equal. 

Q. 9. If m times the m th term of an AP is equal to n times its nth term show that the (m + n)th term of the AP is zero. 

Q. 10.Find the sum of all odd integers between 78 and 500 which are divisible by 7. 

Q. 11. Find n, if the given value of x is nth term of A.P. 17, 22, 27, 32, ...; x = 267 

Q. 12. Find the sum of all the odd numbers between 100 and 200 

Q. 13.If 10 times the 10th term of an AP is equal to 15 times its 15th term, show that its 25th term is zero. 

Q.14. Find the sum 2+4+6+. . . +202 

Q. 15. How many terms are there in the A.P -1,-5/6,-2/3,-1/2……….10/3? Also find its general term? 

Q. 16. 3 times the tenth term is equal to 5 times the twentieth term. Find twentieth term. 

Q. 17.The 5th term of an AP is 24 and its 15th term is 74. Find the sum of its first 10 terms. 

Q. 18. If the first term and last term of an AP are a and l respectively and its sum is S , prove that the common difference of the AP is equal to (l2 – a 2 ) / [2S-(l+a)] . 

Q. 19. If the difference between the 21st and 10th terms of an AP is 55, find the difference between the 45thand 40th terms. 

Q. 20.Find three numbers in an A.P. whose sum is 15 and product 80
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