Animated Solution for Mathematics - Sequence and Series: Let $a_n$ be the $n^{th}$ term of an A. P. If $S_n=a_1+a_2+a_3+\cdots +a_n=700,a_6=7$ and $S_7=7$, then $a_n$ is equal to :

Welcome, future engineers. Today, we are going to dissect a problem that seems simple on the surface but hides a beautiful algebraic structure beneath.

We are dealing with an Arithmetic Progression (AP), the most fundamental sequence in mathematics. To master any AP problem, you must remember that the entire sequence is governed by just two pieces of 'DNA': the first term, denoted as $a$, and the common difference, denoted as $d$.

If you know these two, you know everything about the sequence. Our mission is to uncover these two values using the clues provided.

We are given two powerful pieces of information. First, the sixth term is $a_6=7$.

Using the general formula $a_n=a+(n-1)d$, we can write this as:

This is our first equation. Next, we are told the sum of the first seven terms is $S_7=7$.

The sum formula is $S_n=\frac{n}{2}[2a+(n-1)d]$. Substituting $n=7$, we get:

By dividing both sides by $7$, we simplify this to $\frac{1}{2}(2a+6d)=1$, which elegantly reduces to:

Now, we have a system of two linear equations: $a+5d=7$ and $a+3d=1$. Subtracting the second from the first, we find $2d=6$, which gives us the common difference $d=3$.

Substituting this back, we find $a=-8$. We have successfully decoded the DNA of our sequence!

Now that we have $a=-8$ and $d=3$, we turn to the final part of the puzzle: $S_n=700$. We need to find $n$.

Substituting our known values into the sum formula, we get:

This simplifies to $\frac{n}{2}[-16+3n-3]=700$, or $n(3n-19)=1400$. Expanding this, we arrive at the quadratic equation:

Do not be intimidated by the large constant term. We need to split the middle term $-19n$ into two parts that multiply to $3\times (-1400)=-4200$.

After a moment of calculation, we find $-75$ and $56$. Thus, we factor the equation as:

We have two potential solutions for $n$: $n=-\frac{56}{3}$ and $n=25$. As we discussed, $n$ must be a natural number, so we discard the negative fraction.

We are left with $n=25$. The problem asks for $a_n$, which is now $a_{25}$.

Using our formula $a_{25}=a+24d$, we calculate:

We have arrived at the destination. The answer is 64.

Remember, in JEE Advanced, the math is not just about calculation; it is about the logical flow from the given constraints to the final, elegant result. Keep practicing, and keep falling in love with the process.