Prof. Bhambwani’s
RELIABLE CLASSES /11TH /MATHS-I/ SEQUENCE AND SERIES
Introduction:
A sequence is an arrangement of numbers in a definite order so that there is a definite relation between the numbers and their positions in the arrangement. Each number of the sequence is called a term of the sequence and is denoted by t1, t2, t3…. tn etc.
Definition:
A sequence (tn) is said to be a Arithmetic Progression (A.P.) if tn+1-tn = constant, for all n N.
The constant difference is called the common difference of the A.P. and is denoted by d. Also the first term is denoted by a.
General Tem of an A.P. :
If a is the fist term and d is the common difference for an A.P.(tn), then the nth term is given by tn= a + (n-1) d, n N (verify!)
The sum of the first n terms of an A.P. is given by Sn = (check!)
Note:While solving problems on A.P., it is convenient to take three consecutive terms as a – d, a, a + d. Four consecutive terms as a – 3d, a-d, a+d, a+3d; and five consecutive terms as a – 2d, a – d, a, a+d, a + 2d.
EXERCISE 3.1
1.Find tn for the following A.P.s
(i)4,9,14,19, …………………..(ii)
2.Find
(i)24th term of the A.P. 5, 8, 11, 14 ….
(ii)15th term of the A.P. 21, 16, 11, 6, …….
3.(i)The seventh term of an A.P. is 30 and the tenth term is 21, find the fourth term.
(ii)The third term of an A.P. is -11 and the nineth term is -35, find the nth term the nth term of the A.P.
4.(i)If for a sequence (tn), Sn = 2n2 + 5n, find tn and show that the sequence is an
A.P.
(ii)If for a sequence (tn), Sn = 4n2 – 3n, show that the sequence is an A.P.
5.(i)Find three numbers in A.P. such that their sum is 24 and the sum of their squares is 200.
(ii)Find four numbers in A.P. such that the sum of the first and last number is 8 and the product of second and third number is 12.
6.Find the sum of (i) 1 + 4 + 7 + …… to 22 terms (ii) 3 + 8 + 13 + 18 + ….. to n terms
7.Find the value of n, if
(i)1 + 4 + 7 + 10 + …… to n terms = 590
(ii)50 + 46 + 42 + 38 + …. Ton n terms == 336
8.If for an A.P> (i) S16 = 784, a = 4, find d.
(ii)S12 – 78,d = -3, find a.
9.If for an A.P.
(i) t3 = 17, t7 = 37, find S16.
(ii)t7 = 13, S14 = 203, find S8.
10.Find the sum of all natural numbers from to 200.
(i)which are divisible by 5.
(ii)which are divisible by 3.
You have also studied Geometric Progression. Let us briefly revise it
3.2Geometric Progression (G.P.)
Definition: A sequence {tn} is said to be a geometric progression if constant, for all n N.
The ratio is called the common ratio of the G.P. and is denoted by ‘r’. The first term of the G.P. is denoted by ‘a’.
General term of G.P.
1.If ‘a’ is the first term and ‘r’ is the common ratio of a geometric progression then the nth term is given by arn-1. (verify!)
EXERCISE 3.2
1.For the following G.P.s, find tn.
(i)3, 15, 75, 375,……..
(ii)1, -4, 16, -64, ………..
(iii)1, , 9/4 – 27/8, …………
(iv), , , ………
2.For the G.P.s
(i)if a = 7, r = 1/3, find t6 .
(ii)If a = 5, r= -2, fin t5.
(iii)if a= 2/3, t6 = 162, find r.
(iv)if t8 = 640, r= 2. find a.
3.If for a sequence , , show that the sequence is a G.P. Find its first term and the common ratio.
4.If for a sequence , , show that the sequence is a G.P. Find its first term and the common ratio.
5.(i)Find three numbers in G.P. such that their sum is 35 and their product is 1000.
(ii)Find three numbers in G.P. such that their sum is 13/3 and the sum of their squares is .
6.(i)Find four numbers in G.P. such that their product is 1 and the sum of the middle
two numbers is
(ii)Find five numbers in G.P. such that their product is 32 and the product of fourth and fifth number is 108.
7.If for a given G.P. a = 729 and 7th term is 64, determine S7.
8.Find a G.P. for which sum of the first two terms is -4 and the fifth term is 4 times the third term.
9.If a, b, c, d are in G.P., then prove that
(i)a + b, b+ c, c + d are also in G.P.
(ii)(b – c)2 + (c – a)2 + (d – b)2 = (a – d)2
10.If x, y and z are the pth, qth, rth terms of a G.P. respectively, then show that
xq-r yr-p zp-q = 1
3.3Sum of first n terms of a G.P.:
If ‘a’ is the first term and ‘r’ is the common ratio of a G.P., then verify that the sum Sn, of the first n terms is given by
i)
ii)Sn = na, r = 1
Note: For a G.P.,
i)
EXERCISE 3.3
1.For the following G.P.s. find Sn.
(i)2, 6, 18, 54,…………………
(ii)a, b,
2.For the following G.P.s, find Sn .
(i)2, -2,, 2, -2, ………..
(ii)0.03, 0.06, 0.12, 0.24,………
3.For a G.P.
(i)if a = 1, r = , find S5.
(ii)if S5 = 1023, r = 4, find a.
4.For a G.P.
(i)if a = 2, r = 3, Sn = 242, find n.
(ii)if S3 = 125, S6 = 152, find r.
5.For a G.P.
(i)if t3 = 18, t6 = 486, find S5.
(ii)if t4 = 24, t9 = 768, find S8.
6.Find the sum of n terms
(i)2 +,22 + 222 + …………………
(ii)5 + 55 + 555 + ……………….
7.Find the sum to n terms
(i)0.9 + 0.99 + 0.999 + ……
(ii)0.5 + 0.55 + 0.555 + …..
8.Find the nth terms of the sequence
(i)0.2, 0.22, 0.222, 0.2222, ………
(ii)0.7, 0.77, 0.777, 0.7777, ……..
9.For the sequence, if Sn = 3 (2n – 1), find nth term, hence show that it is G.P. Also find the common ratio.
10.For a sequence, if , find nth term. Hence show that is a G.P. Also find the common ratio.
11.If S1, S2 and S3 are respectively the sum of n, 2n and 3n terms of a G.P. then prove that S1 (S3 – S2) = (S2 – S1)2.
12.If S is the sum, P the product and R the sum of reciprocals of n terms of a G.P. then prove that
3.3.2:Recurring Decimals:
We know that a rational number in the form can be expressed either as a terminating decimal or as a recurring decimal.
For example, is a terminating decimal and is recurring decimal which is briefly written as 0.3.
EXERCISE 3.4
1.Determine whether the sum to infinity of the following G.P.s exist. If it exists, find it.
(i)1,2,4,8,16 ……
(ii), ……………..(iii)1, -3, 9, -27, 81,…..
(iv)1, , ……..(b)5,
2.Express the following recurring decimals as rational number (i.e. in the form ).
(i)(ii)1. (iii)5. (iv) 0.23 (v) 2.3
3.If the first term of a G.P. is 2 and sum to infinity is 6, find the common ratio.
4.The sum of an infinite G.P. is and its common ratio is . Find its first term.
5.The sum of an infinite geometric progression is 15 and the sum of the squares of these is 45. Find the G.P.
EXERCISE 3.5
1.Show that the following sequences are H.P. Also find 10th term in each case.
(i)1/2,1/5,1/8,1/11 ……………
(ii)2/9,1/7,2/19,1/12 ……………..
2.Find the G.M. of two positive numbers whose A.M. and H.M. are 12 and 3 respectively.
3.Find the H.M. of two positive numbers whose A.M. and G.M. are 16 and 8 respectively.
4.Find the G.M. of two positive numbers whose A.M. and H.M. are 9 and 4 respectively.
5.Insert two numbers between 3 and 24 so that the resulting sequence is a G.P.
6.Insert two numbers between -40 and -5 so that the resulting sequence is a G.P.
7.Insert two numbers between 1/4 and 1/13 so that the resulting sequence is a H.P.
8.If the A.M. of two numbers exceeds their G.M. by 10 and their H.M. by 16, find the numbers.
9.Find two numbers whose A.M. exceeds their G.M. by 30 and their H.M. by 48.
10.The A.M. of two numbers exceeds their G.M. by and their G.M. exceeds their H. M. by , find the numbers.
EXERCISE 3.6
Find the following sums (1 to 8)
1.
2.
3.
4.
5.1.3 + 5.7 + 9.11 + ….. upto n terms
6.12 + 32 + 52 + 72 ……… upto n terms
7.(602 - 592) + (582 – 572) + (562 – 552) + ……….. + ((22-12)
8.1.2.3 + 2.3.4 + 3.4.5 + …………. + n (n+ 1) (n+2)
9.If , find n.
10.If S1, S2 and S3 are the sum of first n natural numbers, their squares and their cubes respectively then show that .
3.1Arithmetic Progression (A.P.)
Ex.1.Find tn for the A.P. 32, 28, 24, 20 …………..
Ex.2The tenth term of an A.P. is 1 and the 20th term is -29, find its 3rd term.
Ex.3For an A.P. A = 3, d = 4, find S20
Ex.4Find the value n if 25 + 22 + 19 + …. To n terms = 116
3.2 Geometric Progression (G.P.)
Ex.1For the following G.P.s find the nth term.
(i)2, 6, 18, 54, …….(ii)25, 5, 1, (iii)3,3 3, 3, ……….
Ex.2Show that 3, 6, 12, 24, 48, …. is a G.P. Hence find the 7th term.
Ex.3For a G.P., if a = 5 and , find r and t10.
Ex.4For a G.P., if t3 = 18, t6 = 486, find t9.
Ex.5If for a sequence, tn = show that the sequence is a G.P. Find the First term and the common ratio.
Ex.6Find three numbers in G.P. such that their sum is 21 and the sum of their squares is 189.
Ex.7If the , and terms of a G.P. are a, b and c respectively, then show that .
3.3Sum of first n terms of a G.P.:
Ex.1For G.P. if a = 3 and r = 2, find S10.
Ex.2Find Sn for the following G.P.s
(i)2, 4, 8, 16, ……………….(ii)27, 9, 3, 1, ………………..
(iii)5, -5, 5, -5, ……………….(iv)7, 7, 7, 7, ………………….
Ex.3Find upto n terms.
Ex.4For a G.P. if t3 = 36, t6 = 972 find S4
Ex.5For a G.P. S4 = 81, S8 = 97, find r.
Ex.6If 1 + 4 + 16 + 64 + ………………. Upto n terms = 5461, find n.
Ex.7Find 9 + 99 + 999 + ……. Upto n terms.
Ex.8Find 7 + 77 + 777 + upto n terms.
Ex.9Find the nth term and the sum of first n terms of the sequence 0.3, 0.33, 0.333, 0.3333, ……..
Ex.10For a sequence (tn), if Sn = 5 (2n -1), find tn and show that the sequence is a G.P.
3.3.1The sum to infinity of G.P. :
1.Find Sn and the sum to infinity of the following G.P.s.
(a)
(b)
(c)
3.3.2.Recurring Decimals:
Ex.1 Express the recurring decimal 2.345 as a rational number in the form .
3.4Harmonic Progression (H.P.)
Ex.1Find the nth term of the following H.P. …………………
Ex.2Find the tenth term of the H.P. 1/3, 1/5, 1/7, 1/9, …….
3.5Means
1. Arithmetic Mean (A.M.):
Ex.3Insert n arithmetic means between a and b.
Ex.4Insert 6 Arithmetic means between 3 and 45.
2. Geometric Mean (G.M.)
Ex.5Insert n geometric means a and b.
Ex.6Insert 5 geometric means between 3 and 192
3.Harmonic Mean (H.M)
Ex.1:Find A.M., G.M., and H.M. of the numbers 5 and 45.
Ex.2Insert two numbers between 3 and 81 such that the resulting sequence is a G.P.
Ex.3The A.M. of two numbers exceeds their G.M. by 15 and their H.M. by 27, find the numbers.
3.6 Special series
Ex.1Find
Ex.2Find
Ex.3Evaluate 52 + 62 + 72 + ……..302.
Ex.4Find (502 – 492) + (482 – 472) + (462 – 452) + ……. + (22 – 12)
Ex.5Find
Ex.6Find Sn, if Sn = 1.3 + 2.4 + 3.5 + ……….. upto n terms.
Ex.7If , find n.
Miscellaneous Exercise
Q1.In a G.P. if third term is 63 and the sixth term is 1701, find its nth term.
Q2.For a G.P. if a = 5 and t7 = find r and t9.
Q3.For a sequence, if tn = show that the sequence is a G.P. Find the first term and the common ratio.
Q4.Find three numbers in G.P. such that their product is 216 and the sum of first and third number is 20.
Q5.Find three numbers in G.P. such that their sum is 28 and their product is 512.
Q6.Find four numbers in G.P. such that their product is 729 and the sum of second and third number is 12.
Q7.For a sequence, if Sn = 7(4n – 1),
Find tn and show that the sequence is a G.P.
Q8.The sum of first n terms of a sequence is 5(3n – 1). Show that it is a G.P.
Q9.For two different numbers, if G.M. = 12 and A.M. = 16, find their H.M.
Q10.Find the sum 3 + 33 + 333 + ….. upto n terms.
Q11.Find the sum 8 + 88 + 888 + ….. upto n terms.
Q12.Find the nth term of the sequence 0.5, 0.55, 0.555, …..
Q13.Find the nth term of the sequence 0.4, 0.44, 0.444, 0.4444, ……
Q14.Find
Q15.Find
Q16.Find
Q17.Find
Q18.Find 3.7 + 5.10 + 7.13+9.16+……upto n terms.
Q19.Find 1.4.7+3.6.9+5.8.11+7.10.13+….. upto n terms.
Q20.Find + upto n terms.
Q21.Find 312 + 322 + 332 + …… +502.
Q22.Find (402 – 392) + (382 – 372) + (362 – 352) + ………+(22-12)
Q23.Find the sum
12 . n + 22.(n-1)+32(n-2)+42(n-3)+…… to n terms.
Q24.Find the sum
1+(1 + x ) + (1 + x + x2) + …… + (1 + x + x2 + ….. + xn-1)
Q25.If find n.
Q26.If Find n.
Q27.If Find n.
Q28.If the A.M. of two positive numbers x and y (x > y) is twice their G.M.,
Prove that x : y = ( 2+
Q29.If a, b, c, d are in G.P. prove that (a2 + b2 +c2) (b2 + c2 + d2) = (ab + bc + cd)2
Q30.If a, b, c, d are in G.P. prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
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