Animated Solution for Mathematics - Statistics: The mean of 10 numbers $7\times 8,10\times 10,13\times 12,16\times 14,\dots$ is .

Unlocking the Pattern

Welcome, future engineer! Today, we are going to embark on a journey through a sequence that might look intimidating at first glance, but is actually a beautifully orchestrated pattern.

Imagine you are standing before a series of numbers: $7\times 8,10\times 10,13\times 12,16\times 14,\dots$. At first, it looks like a chaotic jumble of products. But in the world of JEE, chaos is just a pattern waiting to be discovered.

Deconstructing the Sequence

To solve this, we must act like detectives. Look closely at the first factor of each term: $7,10,13,16,\dots$.

Each number is exactly $3$ greater than the one before it. This is a classic Arithmetic Progression (A.P.) with a first term $a=7$ and a common difference $d=3$.

The general term for this, using the formula $a+(n-1)d$, is:

Now, let's turn our attention to the second factor: $8,10,12,14,\dots$. This is another A.P. with a first term $a=8$ and a common difference $d=2$.

Its general term is:

Building the General Term

Now that we have the building blocks, we can construct the general term of our sequence, $T_n$. Since each term is the product of these two factors, we write:

Before we rush into calculations, let's simplify this expression. Expanding the brackets, we get:

This quadratic expression is the DNA of our sequence. Every single term, from the first to the tenth, follows this exact rule.

The Power of Summation

We need the mean of the first $10$ terms. To find the mean, we first need the total sum, $S_{10}={\sum }_{n=1}^{10}{T}_n$.

Using the linearity property of summations, we can distribute the sum across our quadratic expression:

This is where the magic happens. We pull our standard summation formulas from our toolkit: 1. ${\sum }_{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$ 2. ${\sum }_{k=1}^{n}k=\frac{n(n+1)}{2}$ 3. ${\sum }_{k=1}^{n}1=n$

Substituting $n=10$, we get:

Bringing it Home

Watch the elegance of the math as the $6$ in the first term cancels out perfectly! We are left with:

Factoring out $10\times 11$ (which is $110$), we get:

Finally, the mean is simply the sum divided by the number of terms:

See? What started as a daunting sequence became a simple, elegant exercise in pattern recognition and algebraic manipulation. Keep this mindset, and no JEE problem will ever be too big for you to handle!