An arithmetic progression is written in the following way: 2; 5 8; 11 14 17; 20 23 26 29; ... The su

Q: Wait, why is the first term of the $10^{\text{th}}$ row the $46^{\text{th}}$ term? Shouldn't it be the $10^{\text{th}}$ term?

That is a classic trap! The $10^{\text{th}}$ term would only be the start if each row had only one element. But here, the rows grow in size. We have $1$ term in row 1, $2$ in row 2, and so on. Before row 10, we have rows 1 through 9. The total number of terms before row 10 is the sum $1 + 2 + 3 + \dots + 9$, which equals $45$. Therefore, the very next term—the first term of row 10—must be the $46^{\text{th}}$ term of the sequence.

Q: Can I just list out all the numbers until I reach the $10^{\text{th}}$ row?

While you technically could, it is highly inefficient and prone to calculation errors, especially under the time pressure of a JEE Advanced exam. Mastering the algebraic approach—finding the position index and using the sum formula—is the hallmark of a true problem solver. It is faster, cleaner, and much less likely to lead to a 'silly mistake'.

Q: Does the common difference $d$ change when we move from one row to the next?

Great question! No, it does not. The entire triangular arrangement is just a visual way of presenting a single, continuous Arithmetic Progression: $2, 5, 8, 11, \dots$. The common difference $d=3$ is a property of the sequence itself, not the rows. Whether you are looking at row 1 or row 10, the gap between consecutive terms remains constant.

Question

An arithmetic progression is written in the following way: $$\begin{matrix} & & & 2 & & & \ & & 5 & & 8 & & \ & 11 & & 14 & & 17 & \ 20 & & 23 & & 26 & & 29 \ - & - & - & - & - & - & - \ - & - & - & - & - & - & - \end{matrix}$$ The sum of all the terms of the $10^{	ext{th}}$ row is

Sigma Percentile · Accepted Answer

### The Pyramid of Numbers: A Journey into Sequences

Imagine you are standing before a grand, triangular structure built not of stone, but of numbers. This is the beauty of sequences in mathematics—they are not just lists; they are architectures.

Today, we are going to decode the 'Pyramid of Numbers' and uncover the hidden sum of its tenth level.

### Phase 1: Decoding the Structure

Look closely at the arrangement. The first row holds a single value: $2$. The second row holds two values: $5$ and $8$. The third row holds three: $11, 14, 17$.

Do you see the rhythm? The number of terms in any row $n$ is exactly $n$. This is our first geometric insight.

To reach the tenth row, we must first account for all the terms that came before it. We are essentially building a foundation of nine rows. The total number of terms in these first nine rows is the sum of the first nine natural numbers:

$$	ext{Total terms} = 1 + 2 + 3 + \dots + 9 = \frac{9(9+1)}{2} = 45$$

This means that by the time we finish the ninth row, we have already placed $45$ terms. The tenth row, therefore, begins with the $46^{	ext{th}}$ term of our sequence.

### Phase 2: The Hunt for the Starting Point

Now that we know the tenth row starts at the $46^{	ext{th}}$ position, we need to find the value of that term. Our sequence is an Arithmetic Progression (A.P.) starting at $a = 2$ with a common difference of $d = 3$.

The general formula for the $n^{	ext{th}}$ term of an A.P. is our most powerful tool here:

$$T_n = a + (n-1)d$$

Substituting $n = 46$, $a = 2$, and $d = 3$, we calculate:

$$T_{46} = 2 + (46-1)3 = 2 + 45 	imes 3 = 2 + 135 = 137$$

So, the tenth row begins with the number $137$. It is a moment of clarity—the fog lifts, and we see the starting point of our target row.

### Phase 3: The Final Summation

We are now looking at the tenth row in isolation. It is a mini-sequence of $10$ terms, starting with $A = 137$ and maintaining the same common difference $d = 3$.

To find the sum of these ten terms, we use the sum formula for an A.P.:

$$S_n = \frac{n}{2} [2A + (n-1)d]$$

Plugging in our values ($n=10$, $A=137$, $d=3$):

$$S_{10} = \frac{10}{2} [2(137) + (10-1)3]$$

Let us break this down carefully. Inside the brackets, $2 	imes 137 = 274$, and $9 	imes 3 = 27$. Adding these together gives us $301$.

Finally, we multiply by the factor outside, which is $5$:

$$S_{10} = 5 	imes [274 + 27] = 5 	imes 301 = 1505$$

### The Takeaway

We have arrived at **$1505$**. This problem was never really about just adding numbers; it was about understanding the relationship between the position of a term and the structure of the sequence.

Whenever you face a problem like this, do not rush to calculate. Visualize the structure, identify the index of your starting term, and let the elegance of the A.P. formulas guide you to the solution. You have mastered the pyramid!

Animated Solution for Mathematics - Sequence and Series: An arithmetic progression is written in the following way: $20 - - 11 - - 523 - - 214 - - 826 - - 17 - - 29 - -$ The sum of all the terms of the $1 0^{th}$ row is

Enter Numerical Value:

Visualized Solution

Visualizing the Pattern

Identifying the A.P.

Row Structure Analysis

Terms Before Row $10$

First Term of Row $10$

Calculating $T_{46}$

Sum of the $1 0^{th}$ Row

Applying the Sum Formula

Final Calculation

Key Takeaway

The Sigma Insight: Arithmetic Progression (A.P.)

The Pyramid of Numbers

Phase 1

Phase 2

Phase 3

The Takeaway

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