Question 13 of 20advanced🔧 ApplyLong Answer5 marks

Show that 9n+1−8n−99^{n+1} - 8n - 9 is divisible by 6464, whenever nn is a positive integer.

Correct Answer

To show that 9n+1−8n−99^{n+1} - 8n - 9 is divisible by 6464, we need to prove that 9n+1−8n−9=64k9^{n+1} - 8n - 9 = 64k for some natural number kk.

We can write 9n+1=(1+8)n+19^{n+1} = (1 + 8)^{n+1}.

Using the Binomial Theorem:

(1+8)n+1=n+1C0+n+1C1(8)+n+1C2(8)2+…+n+1Cn+1(8)n+1(1 + 8)^{n+1} = {}^{n+1}C_0 + {}^{n+1}C_1(8) + {}^{n+1}C_2(8)^2 + \ldots + {}^{n+1}C_{n+1}(8)^{n+1}

Expanding:

=1+(n+1)(8)+n+1C2(64)+n+1C3(8)3+…+n+1Cn+1(8)n+1= 1 + (n+1)(8) + {}^{n+1}C_2(64) + {}^{n+1}C_3(8)^3 + \ldots + {}^{n+1}C_{n+1}(8)^{n+1} =1+8n+8+64[n+1C2+n+1C3(8)+…+n+1Cn+1(8)n−1]= 1 + 8n + 8 + 64\left[{}^{n+1}C_2 + {}^{n+1}C_3(8) + \ldots + {}^{n+1}C_{n+1}(8)^{n-1}\right] =9+8n+64k= 9 + 8n + 64k

where k=n+1C2+n+1C3(8)+…+n+1Cn+1(8)n−1k = {}^{n+1}C_2 + {}^{n+1}C_3(8) + \ldots + {}^{n+1}C_{n+1}(8)^{n-1} is a natural number.

Therefore:

9n+1−8n−9=64k9^{n+1} - 8n - 9 = 64k

This shows that 9n+1−8n−99^{n+1} - 8n - 9 is divisible by 64 for every positive integer nn.

Exercise: EXERCISE 7.1 | Q: 13 | (Page 8)
For More Understanding

Explanation

This question follows the same pattern as Example 4 in the textbook, which proves divisibility using the Binomial Theorem. The key strategy is to express 99 as (1+8)(1+8), expand using the binomial theorem, and factor out 6464 (=82= 8^2) from all terms except the constant and linear terms.

The constant term is 11 and the linear term coefficient gives 8(n+1)=8n+88(n+1) = 8n + 8. When we subtract 8n8n and 99, all remaining terms contain 6464 as a factor.

Solution Steps

  1. Step 1: Express 9n+1=(1+8)n+19^{n+1} = (1+8)^{n+1}

  2. Step 2: Apply Binomial Theorem: (1+8)n+1=n+1C0+n+1C1(8)+n+1C2(8)2+…(1+8)^{n+1} = {}^{n+1}C_0 + {}^{n+1}C_1(8) + {}^{n+1}C_2(8)^2 + \ldots

  3. Step 3: Expand to get =1+(n+1)(8)+64[n+1C2+…]= 1 + (n+1)(8) + 64[{}^{n+1}C_2 + \ldots]

  4. Step 4: Simplify: =9+8n+64k= 9 + 8n + 64k where kk is a natural number

  5. Step 5: Rearrange: 9n+1−8n−9=64k9^{n+1} - 8n - 9 = 64k, proving divisibility by 6464

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