If an = √(7 + √(7 + √(7 + ....))) having n radical signs then by methods of mathematical induction w

Q: Wait, why can't I just solve for the limit $L = \sqrt{7+L}$?

That is a brilliant question! Solving for the limit $L$ gives you the value the sequence approaches as $n$ goes to infinity, which is $L = \frac{1 + \sqrt{29}}{2} \approx 3.19$. While this tells you the sequence is bounded, the question specifically asks for a proof by Mathematical Induction for all $n \ge 1$. Induction is a rigorous way to prove the bound for every single term, not just the limit.

Q: Why did we choose 7 as the bound? Could we have chosen 4?

You are absolutely right! Since the limit is approximately 3.19, the sequence is indeed bounded by 4 as well. However, in many JEE problems, you are asked to identify which of the given options is true. If $a_n < 7$ is an option, it is mathematically true, even if a tighter bound like $a_n < 4$ also exists. Always check the options provided!

Q: Is induction always the best way for sequences?

Induction is fantastic for proving inequalities and properties of recursive sequences. While other methods like finding the limit or analyzing the function's monotonicity are useful, induction provides a step-by-step logical chain that is often required in formal proofs. It is a fundamental skill in your JEE toolkit.

Question

If $a_n = \sqrt{7 + \sqrt{7 + \sqrt{7 + ....}}}$ having $n$ radical signs then by methods of mathematical induction which is true

Options:

(A) $a_n > 7 \ \forall \ n \ge 1$

(B) $a_n < 7 \ \forall \ n \ge 1$

(C) $a_n < 4 \ \forall \ n \ge 1$

(D) $a_n < 3 \ \forall \ n \ge 1$

Sigma Percentile · Accepted Answer

### Analyzing the Recursive Heart

Let us define our sequence $a_n = \sqrt{7 + \sqrt{7 + \dots}}$ with $n$ radical signs. If you look closely, you will see a beautiful self-similarity.

The term $a_2$ is $\sqrt{7 + \sqrt{7}}$, which is just $\sqrt{7 + a_1}$. Similarly, $a_3$ is $\sqrt{7 + a_2}$.

This reveals the recursive soul of our sequence:
$$a_{n+1} = \sqrt{7 + a_n}$$

This simple relation is the key to everything. We are not dealing with a static number; we are dealing with a dynamic process where each term feeds into the next.

### The Domino Effect

Mathematical Induction is like a row of dominoes. If we can prove that the first domino falls (the base case) and that one falling domino will always knock over the next (the inductive step), then the entire sequence must fall into place.

For our base case, $n=1$, we have $a_1 = \sqrt{7}$. Since $2^2 = 4$ and $3^2 = 9$, we know that $2 < \sqrt{7} < 3$.

Thus, $a_1 < 7$. The first domino has fallen!

### The Inductive Leap

Now, we assume the statement is true for some integer $k$, meaning we assume $a_k < 7$. This is our inductive hypothesis.

Our mission is to prove that $a_{k+1} < 7$. We start with our assumption: $a_k < 7$.

First, add 7 to both sides:
$$7 + a_k < 7 + 7$$
$$7 + a_k < 14$$

Now, take the square root of both sides:
$$\sqrt{7 + a_k} < \sqrt{14}$$

Since $\sqrt{14} \approx 3.74$, and $3.74 < 7$, we have successfully shown that $a_{k+1} < 7$.

### The Grand Conclusion

By the Principle of Mathematical Induction, we have proven that $a_n < 7$ for all $n \ge 1$.

We have taken a seemingly infinite, intimidating expression and broken it down into a logical, step-by-step journey. This is the essence of JEE mathematics: not just finding the answer, but understanding the beautiful, logical structure that makes the answer inevitable.

**The sequence $a_n$ is bounded above by 7 for all $n \ge 1$.** Keep practicing, keep questioning, and never stop loving the process.

Animated Solution for Mathematics - Sequence and Series: If $a_{n} = 7 + 7 + 7 + ....$ having $n$ radical signs then by methods of mathematical induction which is true

Select Answer:

Visualized Solution

The Nested Radical Sequence

Recursive Definition

Graphical Perspective

Base Case ( $n = 1$ )

Evaluating the Base Case

Inductive Hypothesis

Inductive Step Goal

Applying the Recursive Formula

Building the Inequality

Adding 7 to Both Sides

Taking the Square Root

Finalizing the Inductive Step

Conclusion

The Sigma Insight: Sum of Special Series

Analyzing the Recursive Heart

The Domino Effect

The Inductive Leap

The Grand Conclusion

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Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=7+7+7+....​​​ having n radical signs then by methods of mathematical induction which is true