Ex 8.1, 13 - Show that 9n+1 - 8n - 9  is divisible by 64

Ex 8.1,13 - Chapter 8 Class 11 Binomial Theorem - Part 2
Ex 8.1,13 - Chapter 8 Class 11 Binomial Theorem - Part 3 Ex 8.1,13 - Chapter 8 Class 11 Binomial Theorem - Part 4

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 7.1, 13 - Introduction Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer. Numbers divisible by 64 are 64 = 64 × 1 128 = 64 × 2 640 = 64 × 10 Any number divisible by 64 = 64 × Natural number Hence, In order to show that 9n+1 – 8n – 9 is divisible by 64, We have to prove that 9n+1 – 8n – 9 = 64k , where k is some natural number Ex 7.1, 13 Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer. In order to show that 9n+1 – 8n – 9 is divisible by 64, We have to prove that 9n+1 – 8n – 9 = 64k , where k is some natural number Writing (9)n+1 = (1 + 8) n+1 (9)n + 1 = n + 1C0 1(n + 1) + n + 1C1 1(n + 1) – 1 (8)1 + n + 1 C2 1n – 2(8)2 +…… + n + 1Cn + 1 (8) n + 1 We know that (a + b)n = nC0 an + nC1 an – 1 b 1 + nC2 an – 2 b2 + ….. + nCn bn Putting a = 1 ,b = 8 and n = n + 1, (9)n + 1 = n + 1C0 + n + 1C1 8 + n + 1 C2 (8)2 +…… + n + 1Cn + 1 (8) n + 1 = 1 + (𝑛 + 1)!/1!(𝑛 + 1 − 1)! (8) + n + 1 C2(8)2 + n + 1 C3 (8)3 +…… + 1 (8) n + 1 = 1 + ((𝑛 + 1)(𝑛)!)/(𝑛)! (8) + n + 1 C2(8)2 + n + 1 C3 (8)3 +…… + (8) n + 1 = 1 + (n + 1) (8) + n + 1 C2(8)2 + n + 1 C3 (8)3 +…… + (8) n + 1 = 1 + (8n + 8) + n + 1 C2(8)2 + n + 1 C3 (8)3 +…… + (8) n + 1 = 8n + 9 + n + 1 C2(8)2 + n + 1 C3 (8)3 + …… + (8) n + 1 Hence, 9n + 1 = 8n + 9 + n + 1 C2(8)2 + n + 1 C3 (8)3 + …… + (8) n + 1 9n + 1 – 8n – 9 = n + 1 C2(8)2 + n + 1 C3 (8)3 + …… + (8) n + 1 Taking 82 common from right side 9n + 1 – 8n – 9 = (8)2 ("n + 1 C2 + n + 1 C3 (8)3 – 2 + …… + (8) n + 1 – 2" ) 9n + 1 – 8n – 9 = (8)2 ("n + 1 C2 + n + 1 C3 (8)1 + …… + (8) n – 1" ) 9n + 1 – 8n – 9 = 64 ("n + 1 C2 + n + 1 C3 (8)1 + …… + (8) n – 1" ) 9n + 1 – 8n – 9 = 64k where k =("n + 1 C2 + n + 1 C3 (8)1 + …… + (8) n – 1" ) is a natural number Thus , 9n + 1 – 8n – 9 is divisible by 64, Hence proved

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.