Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 7 P (k + 1) is true whenever P(k) is true. Therefore, by principle of mathematical induction, P(n) is true for every positive integer n. We illustrat...
Class XI Mathematics Mathematics 7 first term, (n – 1) + 1 = n in the second term and so on 0 + n = n in the last term. Thus,itcanbeseenthatthesumoftheindicesofaandb isnineverytermof...
Class XI Mathematics Mathematics 7 C x0+ C x + 1 x + ... 2 C x n 3 3 n n n Reprint 2026-27 BINOMIAL THEOREM 131 In particular, for x = 1, we have 2 = C + C0+ C + 1.. + C 2 n n (iii) ...
Class XI Mathematics Mathematics 7 + 5 × 100 × 16 – 32 =10040008000–1000800032=9039207968. Example 3 Which is larger (1.01) 1000000or 10,000? Solution Splitting 1.01 and using binomi...
Class XI Mathematics Mathematics 7 that when divided by 25, 6 – 5nleaves remainder 1. EXERCISE7.1 Expand each of the expressions in Exercises 1 to 5. 5  2 – x  6 1. (1–2x) 2.  x 2...
Class XI Mathematics Mathematics 7 The expansion of a binomial for any positive integral n is given by Binomial Theorem, which is (a + b) = C a + C a n 0 n n 1 n – b + C a 2 n – b + ...
Class XI Mathematics Mathematics 8 Chapter 8 SEQUENCESAND SERIES v Natural numbers are the product of human spirit. – DEDEKIND v 8.1 Introduction In mathematics, the word, “sequence”...
Class XI Mathematics Mathematics 8 a The n term is also called the generalterm of the n. sequence. Thus, the terms of the sequence of person’s ancestors mentioned above are: a 1 2, a...
Class XI Mathematics Mathematics 8 SERIES 137 th In the sequence of primes 2,3,5,7,…, we find that there is no formula for the n prime. Such sequence can only be described by verbal ...
Class XI Mathematics Mathematics 8 9 and 11. n−3 1−3 1 1 (ii) Here a n . Thus, a1= = − ,a =2− ,a =0 3 4 4 2 4 Reprint 2026-27 138 MATHEMATICS 1 1 Hence, the first three terms are – ,...
Class XI Mathematics Mathematics 8 a , a 8. a = n ;a 7 n 17 24 n 2 n(n – 2) 9. a = (–1) n – n ; a 10. a = ;a . n 9 n n+3 20 Reprint 2026-27 SEQUENCES AND SERIES 139 Write the first f...
Class XI Mathematics Mathematics 8 a , a , a , …, a , … is called geometric progression, if each term is 1 2 3 n ak +1 non-zero and a = r (constant), for k ≥ 1. k 2 3 By letting a 1 ...
Class XI Mathematics Mathematics 8 suggests that the n term of a G.P. is given by n–1 a n ar . 2 3 n – 1 2 n – 1 Thus, a, G.P. can be written as a, ar, ar , ar , … ar ; a, ar, ar ,.....
Class XI Mathematics Mathematics 8 = 4. Therefore 131072 = an= 2(4)n – 1 or 65536 = 4n – 1 8 n – 1. Thisgives 4 = 4 So that n – 1 = 8, i.e., n = 9. Hence, 131072 is the 9 term of the...
Class XI Mathematics Mathematics 8 , which gives n = 10. Example 9 The sum of first three terms of a G.P.and their product is – 1. Find the common ratio and the terms. Solution Leta ...
Class XI Mathematics Mathematics 8 number of ancestors preceding the person is 2046. 8.4.3 Geometric Mean (G.M.) The geometric mean of two positive numbers a and b is the number . Th...
Class XI Mathematics Mathematics 8 G.M. Let AandGbeA.M.andG.M.oftwogivenpositiverealnumbersaandb,respectively. Then A= a+b andG = ab Thus, we have A – G = a+b − ab = a+b−2 ab 2 2 ( a...
Class XI Mathematics Mathematics 8 to indicated number of terms in each of the geometric progressions in Exercises 7 to 10: 7. 0.15, 0.015, 0.0015, ... 20 terms. 8. 7 , 21 , 3 7 , .....
Class XI Mathematics Mathematics 8 AR, AR , … AR form a G.P, and find the common ratio. 21. Find four numbers forming a geometric progression in which the third term is greater than ...
Class XI Mathematics Mathematics 8 2026-27 SEQUENCES AND SERIES 147 31. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% com...
Class XI Mathematics Mathematics 8 terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms. 3. The first term...
Class XI Mathematics Mathematics 8 (i) 5 + 55 +555 + … (ii) .6 +. 66 +. 666+… 12. Find the 20 term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms. 13. A farmer buys a used tract...
Class XI Mathematics Mathematics 8 number of terms is called a finite sequence. Asequence is called infinite if it is not a finite sequence. fi Let a 1 a2, 3 , ... be the sequence, th...
Class XI Mathematics Mathematics 9 Chapter 9 STRAIGHTLINES v G eometry, as a logical system, is a means and even the most powerful means to make children feel the strength of the hum...
Class XI Mathematics Mathematics 9 m y 2+ n y1 and (x 2 y2) internally, in the ratio m: n re ,  .  m + n m + n  For example, the coordinates of the point which divides the line ...