JEE Mains · Maths · STD 12 - 1. relation and function
Let \(\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, - \,1),\) where the function \(f\) satisfies \(f(x + y) = f(x) f(y)\) for all natural numbers \(x, y\) and \(f(1) = 2.\) Then the natural number \(‘ a '\) is
- A \(4\)
- B \(16\)
- C \(2\)
- D \(3\)
Answer & Solution
Correct Answer
(D) \(3\)
Step-by-step Solution
Detailed explanation
From the given functional equation: \(f\left( x \right) = {2^x}\forall x \in N\) \({2^{a + 1}} + {2^{a + 2}} + ... + {2^{a + 10}} = 16\left( {{2^{10}} - 1} \right)\) \({2^a}\left( {2 + {2^2} + ... + {2^{10}}} \right) = 16\left( {{2^{10}} - 1} \right)\)…
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