An arithmetic progression is written in the following way, The sum of all the terms of the 10 th row is ______. [2024]

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Solution

Q.
An arithmetic progression is written in the following way

2

5 8

11 14 17

20 23 26 29

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The sum of all the terms of the $10^{th}$ row is ______. [2024]

Ans.
(1505)

First term of the $10^{th}$ row is $10^{th}$ term of sequence 2, 5, 11, 20, 32, ...

$S = 2 + 5 + 11 + 20 + 32 + \dots + T_{10}$

$S =$ $2 + 5 + 11 + 20 + 32 + \dots + T_{9} + T_{10}$

On subtracting, we get

$0 = 2 + \underset{9 terms}{\underset{⏟}{3 + 6 + 9 + 12 + \dots} - T_{10}}$

$\Rightarrow T_{10} = 2 + \underset{9 terms}{\underset{⏟}{3 + 6 + 9 + 12 + \dots}}$

$= 2 + \frac{9}{2} [6 + 8 \times 3] = 2 + 135 = 137$

Terms in $10^{th}$ row with common difference 3 are 137, 140, 143, .....

$S_{10} = \frac{10}{2} [2 \times 137 + 9 \times 3] = 5 \times 301 = 1505$

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