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Chapter 8: Problem 604

If \((666 \ldots \text { n times })^{2}+(8888 \ldots\) n times \()=(4444 \ldots \mathrm{K}\) times \()\) then \(\mathrm{K}=\) (A) \(n+1\) (B) \(\mathrm{n}\) (C) \(2 \mathrm{n}\) (D) \(\mathrm{n}^{2}\)

Short Answer

Expert verified
The short answer is: To find the value of K, compare the simplified sum's expression of \((6^2n + 8^2n)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\) with the given result in terms of 4: \(4(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\). After simplifying, we find that \(K = n\), which corresponds to answer choice \(\boxed{\text{(B) }n}\).

Step by step solution

01

Square the given numbers

We have two numbers: 666... (n times) and 888... (n times). First, we need to square these numbers. The square of 666... (n times) is: \(((6\times10^{(n-1)} + 6\times10^{(n-2)} + \cdots + 6\times10^0))^2\), The square of 888...(n times) is: \(((8\times10^{(n-1)} + 8\times10^{(n-2)} + \cdots + 8\times10^0))^2\).
02

Add the squared numbers

Now, we need to add the squares of these numbers to get the sum: \(((6\times10^{(n-1)} + 6\times10^{(n-2)} + \cdots + 6\times10^0))^2 + ((8\times10^{(n-1)} + 8\times10^{(n-2)} + \cdots + 8\times10^0))^2\).
03

Simplify the expression

Let's simplify the given expression: \((6^2 + 6^2 + \cdots + 6^2)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0) + (8^2 + 8^2 + \cdots + 8^2)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\). Factor out the common term, (10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0): \((6^2 + 6^2 + \cdots + 6^2 + 8^2 + 8^2 + \cdots + 8^2)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\). Combine the like terms: \((6^2n + 8^2n)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\).
04

Define the result in terms of K

Since the result is given as 444...(K times), let's represent it with the number 4: \(4(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\).
05

Find the value of K

Comparing the simplified sum's expression with the given result in terms of 4: \((6^2n + 8^2n)(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0) = 4(10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0)\). Divide both sides by (10^{2(n-1)} + 10^{2(n-2)} + \cdots + 10^0): \(6^2n + 8^2n = 4\). Solve for K: \(K = n\). The correct answer is (B) n.

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Most popular questions from this chapter

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