Animated Solution for Mathematics - Sequence and Series: Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $-\frac{1}{2}$, then the greatest number amongst them is

Imagine you are standing before a complex puzzle. You have five numbers in an Arithmetic Progression (A.P.), their sum is $25$, and their product is $2520$.

At first glance, this looks like a system of equations that could spiral into a nightmare of algebra. But here is the secret that separates the novice from the master: the choice of variables.

When dealing with an odd number of terms in an A.P., never start with $a,a+d,a+2d...$. Instead, embrace symmetry. By defining our terms as $a-2d,a-d,a,a+d,a+2d$, we place the middle term $a$ at the heart of the sequence.

Why does this matter? Because when we sum these terms, the common difference $d$ vanishes like magic:

Just like that, we have anchored our middle term at $a=5$.

Now that we have $a=5$, we turn to the product. We are given:

Substituting $a=5$, we get:

Dividing by $5$, we simplify this to:

Here is where the beauty of algebra shines. If you expand this blindly, you will be lost in a sea of terms. Instead, group the conjugates:

Using the difference of squares identity, $(x-y)(x+y)=x^2-y^2$, this transforms into:

We have tamed the beast!

Expanding this gives us $625-25d^2-100d^2+4d^4=504$. Rearranging, we arrive at the biquadratic equation:

Let $x=d^2$. We are solving $4x^2-125x+121=0$. By splitting the middle term, we find:

This gives us two paths: $d^2=1$ or $d^2=\frac{121}{4}$.

This is the moment of truth. The problem states one of the numbers is $-\frac{1}{2}$. If $d^2=1$, then $d=\pm 1$, and our terms are $3,4,5,6,7$. No $-\frac{1}{2}$ here!

But if $d^2=\frac{121}{4}$, then $d=\pm \frac{11}{2}$. Let's test $d=\frac{11}{2}=5.5$. Our terms become $5-11,5-5.5,5,5+5.5,5+11$, which simplifies to $-6,-0.5,5,10.5,16$.

There it is! The term $-0.5$ is exactly $-\frac{1}{2}$.

We have successfully navigated the constraints. The greatest number in our sequence is:

You see, the math didn't just give us an answer; it told a story of symmetry, reduction, and verification. Keep this mindset, and no A.P. problem will ever intimidate you again. The final answer is 16.