Question

An arithmetic progression is written in the following way

The sum of all the terms of the 10^th row is ______ .

Answer 1

Correct Answer: 1505

Given the sequence:
\(2, 5, 11, 20, \ldots\)

Step 1: Finding the general term
The general term is given by:
\[ T_n = \frac{3n^2 - 3n + 4}{2} \]

Step 2: Finding the 10th term
\[ T_{10} = \frac{3(100) - 3(10) + 4}{2} \] Simplifying:
\[ T_{10} = \frac{300 - 30 + 4}{2} = \frac{274}{2} = 137 \] Hence, \(T_{10} = 137.\)

Step 3: Sum of 10 terms with common difference (c.d.) = 3
\[ S_{10} = \frac{10}{2} \left[ 2(137) + 9(3) \right] \] Simplifying:
\[ S_{10} = 5 (274 + 27) = 5 \times 301 = 1505 \]

Final Answer:
\[ S_{10} = 1505 \]

Answer 2

The sequence given is:

2, 5, 11, 20, …

The general term for the n-th row of this arithmetic progression can be expressed as:

\( T_n = \frac{3n^2 - 3n + 4}{2} \)

For the 10th row, we substitute \( n = 10 \):

\( T_{10} = \frac{3(100) - 3(10) + 4}{2} = \frac{300 - 30 + 4}{2} = \frac{274}{2} = 137 \)

Since there are 10 terms in the 10th row, with a common difference \( c.d. = 3 \), the sum of the terms of the 10th row is given by:

\( \text{Sum} = \frac{10}{2} (2 \times 137 + 9 \times 3) \)

Calculating:

\( \text{Sum} = 5 (274 + 27) = 5 \times 301 = 1505 \)

Answer: 1505