**CBSE NCERT Solutions Chapter 5 Arithmetic Progressions Exercise 5.3 Question 19**

**19. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?**

**Solution:**

The number of logs in the bottom row = 20

The number of logs in the next row = 19

The number of logs in the next to next row = 18

Therefore, we have sequence of the form **20, 19, 18, ....**

First term = a = 20

Common difference = d = 19 - 20 = -1

We need to find that how many rows make total of 200 logs.

**Applying formula**, to find sum of n terms of AP , **we get**

It is a quadratic equation, we can factorize to solve the equation.

We discard n = 25 because we cannot have more than 20 rows in the sequence. The sequence is of the form: 20, 19, 18 ..... At most, we can have 20 or less number of rows.

Therefore, n = 16 which means 16 rows make total number of logs equal to 200.

We also need to find number of logs in the 16th row. In other words, we need to find last term of the sequence.

**Applying formula**, to find sum of n terms of AP , **we get** ( is the last term)

**Therefore, there are 5 logs in the top most row and there are total of 16 rows.**