The number of terms common the two A.P 3, 7, 11, ..., 407 and 2, 9, 16,..., 709 is

Q: Why is the new common difference the LCM of the two original differences?

Think of it as synchronization. If sequence A hits a term every 4 steps and sequence B every 7 steps, they only align when the number of steps taken is a multiple of both 4 and 7. The smallest such interval is the Least Common Multiple, which is 28. Any common term must be 28 units away from the previous one.

Q: Why do we take the minimum of the two last terms as the upper bound?

A common term must exist in *both* sequences. If the first sequence ends at 407, it cannot produce a term like 410, even if the second sequence continues to 709. Therefore, the common sequence is constrained by the shorter of the two.

Q: What if the first common term isn't obvious?

You can always use the general form: $3 + 4(x-1) = 2 + 7(y-1)$. This simplifies to $4x - 1 = 7y - 5$, or $4x = 7y - 4$. You can test small integer values for $y$ until $7y-4$ is divisible by 4. It's a robust algebraic way to find the anchor point.

Question

The number of terms common the two A.P $3, 7, 11, \dots, 407$ and $2, 9, 16,\dots, 709$ is

Sigma Percentile · Accepted Answer

### The Rhythm of Numbers: A Journey into Intersecting Progressions

Welcome, fellow traveler on the path to JEE mastery! Today, we are not just solving a problem; we are uncovering the hidden rhythm of numbers.

Imagine two runners on a track. One takes long, steady strides, while the other takes shorter, quicker ones. We want to know how many times they land on the exact same spot.

This is the essence of finding common terms in two arithmetic progressions (A.P.s). Let's break this down with the precision of a mathematician and the heart of a mentor.

### Phase 1: The Anatomy of the Sequences

First, let's look at our two sequences. The first A.P., let's call it $A_1$, is $3, 7, 11, \dots, 407$.

Here, the first term $a_1 = 3$, and the common difference $d_1 = 7 - 3 = 4$. It is a steady climb.

The second A.P., $A_2$, is $2, 9, 16, \dots, 709$. Here, $a_2 = 2$, and the common difference $d_2 = 9 - 2 = 7$. These sequences are like two different musical beats playing simultaneously.

### Phase 2: The Hunt for the Anchor

To find the common terms, we need a starting point—an anchor. Let's list the first few terms of each.

For $A_1$, we have $3, 7, 11, 15, 19, 23, \dots$. For $A_2$, we have $2, 9, 16, 23, 30, \dots$.

Look closely! At the number $23$, the two sequences finally meet. This is our first common term, $a_c = 23$. This is the moment of synchronization.

### Phase 3: The LCM Magic

Now, here is the beautiful, elegant property of arithmetic progressions. The common terms of two A.P.s will always form a new A.P. of their own!

But what is the step size of this new sequence? It is the Least Common Multiple (LCM) of the original differences.

We have $d_1 = 4$ and $d_2 = 7$. The $	ext{LCM}(4, 7) = 28$. This means that after our first common term of $23$, the next common term will appear exactly $28$ steps later.

The new sequence is $23, 51, 79, \dots$. Isn't that satisfying?

### Phase 4: The Boundary Constraint

We cannot go on forever. A common term must exist in *both* original sequences.

Therefore, it cannot exceed the smaller of the two last terms. We compare $407$ and $709$.

The minimum is $407$. So, our common terms must satisfy $T_n \le 407$. This is our strict upper bound.

### Phase 5: The Final Inequality

The general term of our new common A.P. is $T_n = a_c + (n-1)d_c$. Substituting our values, we get:

$$23 + (n-1)28 \le 407$$

Now, let's solve for $n$. Subtracting $23$ from both sides gives $(n-1)28 \le 384$.

Dividing by $28$, we get:

$$n-1 \le \frac{384}{28} \approx 13.71$$

Adding $1$ to both sides, we find $n \le 14.71$. Since $n$ must be a positive integer, the largest possible value is $14$.

### Conclusion

There you have it! There are exactly **$14$** common terms.

The beauty of this problem lies in the realization that complex intersections can be reduced to simple, elegant rules. Whenever you face two sequences, look for the LCM of their differences, find that first anchor point, and let the inequality guide you to the answer.

You have the tools; now go forth and conquer!

Animated Solution for Mathematics - Sequence and Series: The number of terms common the two A.P $3, 7, 11, \dots, 407$ and $2, 9, 16, \dots, 709$ is

Enter Numerical Value:

Visualized Solution

Analyze the First A.P.

Analyze the Second A.P.

Find the First Common Term

The LCM Rule for Common Difference

Determine the Upper Bound

Set up the Inequality

Isolate the Term with $n$

Divide by the Common Difference

Final Calculation for $n$

Conclusion and Key Takeaway

The Sigma Insight: Arithmetic Progression (A.P.)

The Rhythm of Numbers

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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; } The number of terms common the two A.P 3,7,11,…,407 and 2,9,16,…,709 is