Let n ≥ 5 be an integer. If 9n – 8n – 1 = 64α and 6n – 5n – 1 = 25β, then α – β is equal to
Let n ≥ 5 be an integer. If 9n – 8n – 1 = 64α and 6n – 5n – 1 = 25β, then α – β is equal to
\(1 + ^nC_2 (8-5) + ^nC_3(8² - 5²) + ... + ^nC_n ( 8^{n-1} - 5^{n-1} )\)
\(1 + ^nC_3 (8-5) + ^nC_4(8² - 5²) + ... + ^nC_n ( 8^{n-2} - 5^{n-2} )\)
\(^nC_3 (8-5) + ^nC_4 ( 8²-5²) + ... + ^nC_n ( 8^{n-2} - 5^{n-2} )\)
\(^nC_4 (8-5) + ^nC_5 ( 8²-5²) + ... + ^nC_n ( 8^{n-3} - 5^{n-3} )\)
The Correct Option is C
Solution and Explanation
The correct answer is (C) : \(^nC_3 (8-5) + ^nC_4 ( 8²-5²) + ... + ^nC_n ( 8^{n-2} - 5^{n-2} )\)
(1 + 8)n – 8n – 1 = 64α
\(⇒ 1 + 8n + ^nC_28^2 + ^nC_38^3 + ..... +^nC_n8^n - 8n-1 = 64α\)
\(⇒ α = ^nC_2 + ^nC_38 + ^nC_48^2 + ..... + ^nC_n8^{n-2} ...... (i)\)
Similarly
(1+5)n - 5n-1=25β
\(⇒ 1 + 5n + ^nC_25^2 + ^nC_35^3 + ..... + ^nC_n5^n - 5n - 1 = 25β\)
\(⇒ β = ^nC_2 + ^nC_3.5 + ^nC_4.5^2 + ...... + ^nC_n 5^{n-2} ..... (ii)\)
\(α - β = ^nC_3(8-5) + ^nC_4 (8^2-5^2) + ..... + ^nC_n(8^{n-2} - 5^{n-2})\)
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.