Let x₁, x₂, x₃, ..., x₂₀ be in geometric progression with x₁ = 3 and the common ratio 1/2. A new dat

Q: Wait, why do we ignore the $(1/2)^{20}$ term? Isn't that mathematically imprecise?

In JEE Advanced, precision is vital, but so is efficiency. The term $(1/2)^{20}$ is approximately $10^{-6}$. When multiplied by constants like 4 or 12, it remains incredibly small. Since we only need the greatest integer (the floor), this negligible value will not change the integer part of our result. Always check the scale of your error before discarding terms!

Q: Is there a formula for the AGP sum, or do I have to derive it every time?

While there are general formulas for AGP sums, deriving it via the 'shift and subtract' method is much safer under exam pressure. It builds intuition and prevents memory-based errors. It is a robust, algorithmic approach that works every time.

Q: Why can't I just calculate each $y_i$ and sum them up?

Calculating twenty individual terms would be tedious and highly prone to arithmetic errors. The beauty of this problem lies in identifying the underlying patterns. By using summation properties, you reduce a massive calculation to a few elegant, standard formulas.

Question

Let $x_1, x_2, x_3, \dots, x_{20}$ be in geometric progression with $x_1 = 3$ and the common ratio $\frac{1}{2}$. A new data is constructed replacing each $x_i$ by $(x_i - i)^2$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is ________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the edge of a complex mathematical landscape. You are given a geometric progression, a sequence that grows or decays with rhythmic precision.

We start with $x_1 = 3$ and a common ratio $r = \frac{1}{2}$. Our goal is to understand a new dataset, $y_i = (x_i - i)^2$, and find the mean of this transformation for $n=20$.

### Breaking Down the Beast

The expression $(x_i - i)^2$ looks intimidating at first glance. However, in mathematics, when you face a complex expression, your first instinct should be to simplify.

By expanding the square, we get:
$$(x_i - i)^2 = x_i^2 + i^2 - 2ix_i$$

This is the turning point. We have transformed one complex summation into three distinct, manageable pillars: $\sum x_i^2$, $\sum i^2$, and $\sum ix_i$.

### The Three Pillars

First, we tackle $\sum x_i^2$. Since $x_i = 3(\frac{1}{2})^{i-1}$, squaring it gives $9(\frac{1}{4})^{i-1}$.

This is a new geometric progression with a first term of $9$ and a common ratio of $\frac{1}{4}$. Using the sum formula $S = a \frac{1-r^n}{1-r}$, we find this sum is approximately:
$$S_1 = 9 \left( \frac{1 - (1/4)^{20}}{1 - 1/4} \right) \approx 12$$

Second, we have $\sum i^2$, the sum of the squares of the first $20$ natural numbers. This is a classic standard result given by the formula:
$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$

Plugging in $n=20$, we get:
$$\frac{20(21)(41)}{6} = 2870$$

### The AGP Masterclass

Finally, we face the most interesting challenge: the Arithmetico-Geometric Progression (AGP) $\sum ix_i$. The term $\sum ix_i$ is a beautiful dance between an arithmetic progression ($i$) and a geometric progression ($x_i$).

To solve this, we use the 'shift and subtract' method. We write the sum $S$, multiply it by the common ratio $\frac{1}{2}$, and shift the terms.

When we subtract these two, the middle terms collapse into a simple geometric series. This is the elegance of algebra—turning a complex, multi-layered problem into a simple, solvable one. After the calculation, we find this sum is approximately $12$.

### The Final Assembly

Now, we bring it all together. The mean $\bar{y}$ is defined as:
$$\bar{y} = \frac{1}{20} \left[ \sum_{i=1}^{20} x_i^2 + \sum_{i=1}^{20} i^2 - 2 \sum_{i=1}^{20} ix_i \right]$$

Substituting our values, we get:
$$\bar{y} = \frac{1}{20} [12 + 2870 - 2(12)] = \frac{2858}{20} = 142.9$$

The final step is to find the greatest integer less than or equal to $142.9$. The final answer is **142**.

Enter Numerical Value:

Visualized Solution

Define the Geometric Progression $x_{i}$

Define the Transformed Data $y_{i}$

Expand the Squared Term

Calculate $\sum x_{i}^{2}$

Calculate $\sum i^{2}$

Calculate $\sum i x_{i}$

Evaluate the AGP Sum

Combine Sums to Find Mean $\overset{x}{ˉ}$

Final Result

The Sigma Insight: Arithmetic-Geometric Progression (A.G.P.)

Analyzing the Setup

Breaking Down the Beast

The Three Pillars

The AGP Masterclass

The Final Assembly

Similar Questions

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If $(10)^{9} + 2 (11)^{1} (10)^{8} + 3 (11)^{2} (10)^{7} + \dots + 10 (11)^{9} = k (10)^{9}$ , then $k$ is equal to:

Enter Numerical Value:

Visualized Solution

Define the Geometric Progression .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xi​

Define the Transformed Data .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } yi​

Expand the Squared Term

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑xi2​

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑i2

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∑ixi​