Question:
For each natural number $n, (n + 1)^7 - n^7 -1$ is divisible by 7.
For each natural number $n, n^7 - n$ is divisible by 7.
For each natural number $n, (n + 1)^7 - n^7 -1$ is divisible by 7.
For each natural number $n, n^7 - n$ is divisible by 7.
Updated On: Mar 11, 2024
- Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
- Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
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The Correct Option is A
Solution and Explanation
The correct answer is A:Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
Given that;
Statement(S1): For each natural number ‘n’ \((n+1)^7-n^7-1\) is divisible by 7
Statement(S2): For each natural number n, \(n^7-n\) is divisible by 7.
Let us use mathematical induction that can check statement 2 is true for \(\forall n\in N\)
\(\therefore (n+1)^7-n^7-1=[(n+1)^7-(n+1)]-[n^7-n]\)
Here both the terms are divisible by 7
\(\therefore(n+1)^7-n^7-1\) is also divisible by 7
So, both these statements are true.
Given that;
Statement(S1): For each natural number ‘n’ \((n+1)^7-n^7-1\) is divisible by 7
Statement(S2): For each natural number n, \(n^7-n\) is divisible by 7.
Let us use mathematical induction that can check statement 2 is true for \(\forall n\in N\)
\(\therefore (n+1)^7-n^7-1=[(n+1)^7-(n+1)]-[n^7-n]\)
Here both the terms are divisible by 7
\(\therefore(n+1)^7-n^7-1\) is also divisible by 7
So, both these statements are true.
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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.