1. Key terms:
Arithmetic progression (A.P.) : An A.P. is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference of the A.P.
If a is first term and d is common difference of an A.P. , then the A.P is a , a +d , a+2d , 2+3d …..
The term of an a.p is denoted by and = a+(n1) d , where a = first term and d = common difference.
term from the end = l – (n1) d , where l = last term.
Three terms ad , a , a+d are in A.P with common difference d.
Four terms a3d , ad , a+d ,a+3d are in A.P with common diff. 2d .
The sum of first n natural number is
The sum of n terms of an A.P with first term a and common difference d is denoted by = { 2a+(n1) d }
also , = (a+l) where , l = last term.
=  . Where = term of an A.P
D =  . Where d = common difference of an A.P.
2. Let us denote the first term of an AP by a1, second term by a2 , . . ., nth term by an and the common difference by d. Then the AP becomes a1, a2, a3, . . ., an. Þ So, a2 – a1 = a3 – a2 = . . . = an – an – 1 = d.
Or, the common difference = d = ak + 1 – ak is the same every time.
A particular sequence is an AP or not by checking a2 – a1 = a3 – a2
3. Finding nth tern of AP
a1 = a
a2 = a +d = a + [21]d
a3 = a +2d = a + [31]d

an = a + [n1]d
nth term : an = a + [n1]d
OR, nth term = l = last term of AP
Also,
4. Sum of First n Terms of an AP
Let us denote the first term of an AP by a1, second term by a2 , . . ., nth term by an and the common difference by d. Then the AP becomes a1, a2, a3. . . a + [n1]d
Let S denote the sum of the first n terms of the AP. We have
S = a + (a + d ) + (a + 2d ) + . . . + [a + (n – 1) d ] (i)
Rewriting the terms in reverse order, we have
S = [a + (n – 1) d ] + [a + (n – 2) d ] + . . . + (a + d ) + a (ii)
On adding (i) and (ii), termwise. we get
2S = [2a + (n – 1) d ] + [2a + (n – 1) d ] + . . . + [2a + (n – 1) d ]  n times
S = [2a + (n – 1) d ]
OR, S = [a + a + (n – 1) d ] = [a + an ]
OR, if there are only n terms in an AP, then an = l, the last term.
S = (a + l )
This form of the result is useful when the first and the last terms of an AP are given and the common difference is not given.
5. The nth term of an AP is the difference of the sum to first n terms and the sum to first (n – 1) terms of it, i.e.,
an = Sn – Sn – 1.
VALUE BASED QUESTION
1. Ram asks the labour to dig a well upto a depth of 10 metre. Labour charges Rs. 150 for first metre and Rs. 50 for each subsequent metres. As labour was uneducated, he claims Rs. 550 for the whole work. What should be the actual amount to be paid to the labours? What value of Ram is depicted in the question if he pays Rs. 600 to the labour?
2. Nidhi saves Rs. 2 on first day of the month, Rs. 4 on second day, Rs. 6 on third day and so on. What will be her saving in the month of Feb. 2012? What value is depicted by Nidhi?
3. 200 logs are stacked such that 20 logs are in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed? What value is depicted in the pattern of logs?
4. How many two digit numbers are there in between 6 and 102 which are divisible by 6. Ram calculated it by using A.P. while Shyam calculated it directly. Which value is depicted by Ram?
5. In a school, students thought of planting trees in an around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying e.g. a section of classI will plant 1 tree, a section of class II will plant 2 trees and so on till class XII. There are three sections of each class. How many trees will be planted by the students? What value can you infer from the planting the trees?
6. Rs. 9000 were divided equally among a certain number of students. Amit was given the responsibility of dividing this amount among the students but 20 more students admitted to the school. Now each student got Rs. 160 less. Find the original number of students? What value of Amit is depicted in the question?
Ans:
1. Rs. 600, Honesty, Sincerity.
2. Rs. 870, Economy, Saving
3. 16, Space saving, Creative, Reasoning, Balancing
4. 15, Time Saving, Seasoning
5. 234, Environmental, Social
6. 25, Logical, Sincerity, Leadership
Practice Question carry 2 marks each. [10th Class – Maths chapter Arithmetic progressions]
Q. Which term of the A.P : 120, 116, 112, ….. is its first negative term.
Q. If k+1, 3k and 4k+2 be any three consecutive terms of an A.P., find the value of k.
Q. Determine k so that k +2, 4k  6 and 3k  2 are the three consecutive terms of an A.P.
Q. The sum of 5th and 7th term of an A.P. is 52 and the 10th term is 46. Find the common difference.
Q. Is 68 a term of the A.P : 7, 10, 13, . . . . . . . . ?
Q. Which term of A.P. : 21, 18, 15, …….. is zero ?
Q. Which term of the A.P. 45, 41, 37, 33,…… is the first negative term ?
Q. If 3x + k, 2x + 9 and x+13 are three consecutive terms of an A.P., find k.
Q. Find the middle term of the A.P. 11, 7,  3, ……. , 45
Q. Find the number of terms in the A.P. 18, , 13, ……..,  47
Q. Find the sum of first 15 terms of an A.P. whose nth term is 3 2n.
Q. If the numbers x  2, 4 x  1,& 5 x + 2 are in A.P., then find the value of x
Q. Find the 10th term from the end to beginning of the AP : 4, 9, 14, ……….., 254.
Q. Which term of A.P : 17, 14, 11,  is  43 ?
Q. Is 184 a term of the sequence 3, 7, 11 …… ?
Q. Find the 7th term from the end of the A.P 7, 10, 13 ….. 184.
Q. how many two digit number are divisible by 3 [CBSE 2012]
Q. Find the sum of all three digit natural number , which are multiple of 11
Q. In an AP , the first term is 12 and the common difference is 6. if the last term of AP . is 252, find its middle term.
Q. If the number X 2 , 4X 1 and 5X+2 are in AP. , Find value of X
Q. If the nth term of an AP is (2n+1), find the sum of first n terms of the AP [Ans. Sn=n(n+2)]
Q. The ratio of the sums of m and n terms of an AP is : .show that the ratio of the mth and nth terms is (2m1) : (2n1).
Q. In an AP, the sum of first n terms is , Find it 25th term.
Practice Question carry 2 marks each. [10th Class – Maths chapter Arithmetic progressions]
Q. Which term of the A.P : 120, 116, 112, ….. is its first negative term.
Q. If k+1, 3k and 4k+2 be any three consecutive terms of an A.P., find the value of k.
Q. Determine k so that k +2, 4k  6 and 3k  2 are the three consecutive terms of an A.P.
Q. The sum of 5th and 7th term of an A.P. is 52 and the 10th term is 46. Find the common difference.
Q. Is 68 a term of the A.P : 7, 10, 13, . . . . . . . . ?
Q. Which term of A.P. : 21, 18, 15, …….. is zero ?
Q. Which term of the A.P. 45, 41, 37, 33,…… is the first negative term ?
Q. If 3x + k, 2x + 9 and x+13 are three consecutive terms of an A.P., find k.
Q. Find the middle term of the A.P. 11, 7,  3, ……. , 45
Q. Find the number of terms in the A.P. 18, , 13, ……..,  47
Q. Find the sum of first 15 terms of an A.P. whose nth term is 3 2n.
Q. If the numbers x  2, 4 x  1,& 5 x + 2 are in A.P., then find the value of x
Q. Find the 10th term from the end to beginning of the AP : 4, 9, 14, ……….., 254.
Q. Which term of A.P : 17, 14, 11,  is  43 ?
Q. Is 184 a term of the sequence 3, 7, 11 …… ?
Q. Find the 7th term from the end of the A.P 7, 10, 13 ….. 184.
Q. how many two digit number are divisible by 3 [CBSE 2012]
Q. Find the sum of all three digit natural number , which are multiple of 11
Q. In an AP , the first term is 12 and the common difference is 6. if the last term of AP . is 252, find its middle term.
Q. If the number X 2 , 4X 1 and 5X+2 are in AP. , Find value of X
Q. If the nth term of an AP is (2n+1), find the sum of first n terms of the AP [Ans. Sn=n(n+2)]
Q. The ratio of the sums of m and n terms of an AP is : .show that the ratio of the mth and nth terms is (2m1) : (2n1).
Q. In an AP, the sum of first n terms is , Find it 25th term.
Question carry 3 marks each. Click Here Arithmetic Progressions
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