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The value of 42+42+42+тАжтАЛтАЛтАЛ is:
6
7
8
9
7
For expressions of type N+N+тАжтАЛтАЛ, split N into two consecutive integers n(n+1). The answer is the larger number (n+1) for a plus sign and the smaller number n for a minus sign.
An infinite nested radical of the form x = sqrt(42 + sqrt(42 + sqrt(42 + ...)))
x=N+N+тАжтАЛтАЛтЯ╣x2тИТxтИТN=0
For expressions of type N+N+тАжтАЛтАЛ, split N into two consecutive integers n(n+1). The answer is the larger number (n+1) for a plus sign and the smaller number n for a minus sign.
Many students try to solve it by squaring the entire expression multiple times or guessing wrong factors like 3 and 14 instead of consecutive integers 6 and 7.
Set up the equation
Let the given expression be x. Since it is an infinite series, the term inside the first square root is also x.
x=42+xтАЛ
Square both sides
To remove the outer square root, square both sides of the equation.
x2=42+x
Form a quadratic equation
Rearrange the terms to form a standard quadratic equation ax2+bx+c=0.
x2тИТxтИТ42=0
Solve by factoring
Factor the quadratic equation by finding two numbers that multiply to -42 and add to -1. The factors are -7 and 6.
(xтИТ7)(x+6)=0
Find the valid value
The roots are x=7 and x=тИТ6. Since the square root of a positive number must be positive, we discard x=тИТ6.
x=7
B is correct because the infinite nested radical evaluates to 7 when solved as a quadratic equation.
This concept is a classic in competitive exams; always check if the number N can be expressed as a product of two consecutive integers n(n+1) to find the answer in seconds.
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