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 (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?
Fig. 5.5

Question

For positive integers $ n $, if $ 4 a_n = \frac{n^2 + 5n + 6}{4} $ and $$ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} $$

Answer 1

The number of logs in each row forms an arithmetic progression (A.P.): $20, 19, 18, \dots$. For this A.P., the first term is $a = 20$ and the common difference is $d = a_2 - a_1 = 19 - 20 = -1$. If a total of 200 logs are placed in $n$ rows, then the sum of logs in $n$ rows, $S_n$, is 200.

The formula for the sum of an A.P. is $S_n = \frac n2[2a + (n-1)d]$.

Substituting the given values: $200 = \frac n2[2(20) + (n-1)(-1)]$. Multiplying both sides by 2 gives $400 = n[40 - (n-1)]$, which simplifies to $400 = n(40 - n + 1)$ or $400 = n(41 - n)$. Expanding this equation results in $400 = 41n - n^2$. Rearranging into a quadratic equation gives $n^2 - 41n + 400 = 0$. Factoring the quadratic equation yields $n^2 - 16n - 25n + 400 = 0$, which further factors to $n(n - 16) - 25(n - 16) = 0$, and finally $(n - 16)(n - 25) = 0$. Therefore, the possible values for $n$ are $n = 16$ or $n = 25$.

To determine the number of logs in these rows, we use the formula for the $n$-th term of an A.P.: $a_n = a + (n - 1)d$. For $n = 16$, $a_{16} = 20 + (16 - 1)(-1) = 20 - 15 = 5$. For $n = 25$, $a_{25} = 20 + (25 - 1)(-1) = 20 - 24 = -4$. Since the number of logs in a row cannot be negative, the 25th row scenario is not possible.

Consequently, 200 logs can be arranged in 16 rows, with the 16th row containing 5 logs.

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 (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?
Fig. 5.5

Solution and Explanation

Top Questions on Sum of First n Terms of an AP

Questions Asked in CBSE Class X exam

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 (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?Fig. 5.5

Solution and Explanation

Top Questions on Sum of First n Terms of an AP

Questions Asked in CBSE Class X exam

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 (see Fig. 5.5). In how many rows are the 200 logs placed and how many logs are in the top row?
Fig. 5.5