Question

\(\mathrm{S}_{\mathrm{n}}=2 \cdot 7^{\mathrm{n}}+3.5^{\mathrm{n}}-5, \mathrm{n} \in \mathrm{N}\) is divisible by the multiple of (a) 5 (b) 7 (c) 24 (d) None of these

Short answer

The sequence \(S_n = 2 \cdot 7^n + 3 \cdot 5^n - 5\) is divisible by 24 for odd values of n and divisible by 7 for even values of n. Therefore, there is no fixed multiple that it is always divisible by, and the answer is (d) None of these.

Step-by-step solution

Calculate Sn for different values of n to find the pattern

For n=1, 2, 3, we have: \(S_1 = 2 \cdot 7^1 + 3 \cdot 5^1 - 5 = 14 + 15 - 5 = 24\) \(S_2 = 2 \cdot 7^2 + 3 \cdot 5^2 - 5 = 98 + 75 - 5 = 168\) \(S_3 = 2 \cdot 7^3 + 3 \cdot 5^3 - 5 = 686 + 375 - 5 = 1056\)

Check divisibility

Use the values of Sn to check for divisibility: 24/5 = 4.8 (Not divisible by 5) 24/7 = 3.42 (Not divisible by 7) 24/24 = 1 (Divisible by 24) 168/5 = 33.6 (Not divisible by 5) 168/7 = 24 (Divisible by 7) 168/24 = 7 (Not divisible by 24) 1056/5 = 211.2 (Not divisible by 5) 1056/7 = 50.8 (Not divisible by 7) 1056/24 = 44 (Divisible by 24)

Identify the pattern

We notice from Steps 1 and 2 that Sn is divisible by 24 for odd values of n and divisible by 7 for even values of n.

Conclusion

From Step 3, we can conclude that there is no fixed multiple that Sn is always divisible by. Hence, the answer is (d) None of these.

Key concepts

Sequence and Series

Understanding the properties of sequences and series is fundamental to solving problems that involve patterns of numbers. In the given exercise, the term \(S_n\) represents a sequence where each term is governed by a specific formula: \(S_n = 2 \cdot 7^n + 3 \cdot 5^n - 5\). A sequence is simply a set of numbers in a specific order. When we talk about series in mathematics, we refer to the sum of the terms of a sequence. The distinction is important because a series is the accumulated total when the terms of the sequence are added together.

By evaluating the sequence for the first few terms (\(n=1, 2, 3\)), we can search for patterns or properties that might help us deduce the problem's divisibility rule. Notably, in this instance, it wouldn't be practical to use a series approach since we aren't adding the terms together; we're analyzing each term independently for divisibility.

In our specific case, calculating the first few terms helps reveal whether there is a consistent multiple that each term of the sequence is divisible by. As sequences can sometimes exhibit complex behaviors, this exploratory step is crucial to spot any predictable trends or mathematical nuance within the sequence.

Mathematical Induction

Mathematical induction is a proof technique particularly useful for proving the properties of a sequence, like the divisibility of its terms as seen in the exercise. However, our current step-by-step solution does not employ mathematical induction. Normally, mathematical induction helps establish the truth of a statement for all natural numbers. It comprises two key steps: the base case, which verifies the statement for the initial value (usually \(n=1\)), and the inductive step, which assumes the statement is true for some \(n\) and then proves it for \(n+1\).

In the context of our exercise, if we were to approach it using induction, we would prove the divisibility for the first term \(S_1\) first, assume divisibility for \(S_k\), and then attempt to prove it for \(S_{k+1}\). However, as we saw in the solution, the pattern of divisibility changes depending on whether \(n\) is odd or even, so a straightforward inductive proof would not apply here, and a different inductive argument would be needed for each case. This makes the inductive approach less suited for this problem.

Number Theory

Number Theory is a branch of mathematics concerned with the properties and relationships of integers. It includes the study of prime numbers, divisibility rules, and modular arithmetic, among other topics. For the exercise provided, knowledge of number theory enables us to understand and apply divisibility rules. Divisibility rules help us quickly determine if one number is a multiple of another without performing long division.

Using number theory, we look at the structure of \(S_n\) for patterns in prime factorization, which might indicate a common divisor. However, the mixed powers of 7 and 5 in combination with the subtraction of 5 make it challenging to discern a straightforward rule that would apply to all terms in the sequence. The step-by-step solution confirms this as it shows that no single divisor applies to all terms, leading to the conclusion that there is no fixed multiple that \(S_n\) is always divisible by. By utilizing number theory and exploring the properties of the numbers involved, one can work towards a logical conclusion, as demonstrated in the step-by-step solution.