Animated Solution for Mathematics - Sequence and Series: If $a_1,a_2,a_3,\dots$ are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first 15 terms of this A.P. is :

Welcome, future engineers. Today, we are going to dismantle a beautiful problem from the world of Arithmetic Progressions.

Often, when students see a problem like this, their first instinct is to panic. They see $a_1+a_7+a_{16}=40$ and immediately think, "I have two variables, $a$ and $d$, but only one equation! How can I possibly solve this?"

This is the exact moment where the JEE tests your maturity. It is not testing your ability to solve a system of equations; it is testing your ability to see the structure of the math.

Let us start by laying out our tools. We know that for any Arithmetic Progression, the $n$-th term is defined by the formula $a_n=a+(n-1)d$, where $a$ is the first term and $d$ is the common difference.

Our given clue is $a_1+a_7+a_{16}=40$. Let us translate this into our algebraic language:

When we sum these up, we get:

Look at that equation: $3a+21d=40$. Do you see the hidden potential? Both $3$ and $21$ are divisible by $3$.

If we divide the entire equation by $3$, we get:

This, my friends, is our "Golden Key." We do not know $a$, and we do not know $d$, but we know exactly what their specific combination, $a+7d$, equals. In the high-stakes environment of the JEE, this is a massive win.

The Bridge to the Answer

Now, let us look at what the problem actually asks for: the sum of the first $15$ terms, $S_{15}$. The standard formula for the sum of the first $n$ terms is:

Substituting $n=15$ into this formula gives us:

Do you see the beauty of the structure? Inside that bracket, we have $2a+14d$. If we factor out a $2$, we get $2(a+7d)$.

This is not a coincidence; it is the mathematical design of the problem. Watch what happens when we substitute this back into our sum expression:

The $2$ in the numerator and the $2$ in the denominator cancel out perfectly, leaving us with:

We are now at the finish line. We already found our Golden Key, $a+7d=\frac{40}{3}$. All that remains is a simple substitution:

Dividing $15$ by $3$ gives us $5$, and $5\times 40$ is $200$. The sum of the first $15$ terms is $200$.

Think about what we just did. We never found $a$. We never found $d$.

We simply respected the structure of the A.P. and let the algebra guide us to the answer. This is the mindset of a topper. When you face a problem, do not just start calculating; pause, look for the symmetry, and find the Golden Key.