Math, asked by Dhdfr, 8 months ago

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 pic). In how many rows are the 200 logs placed and how many logs are in the top row?

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Answered by Anonymous

$\boxed{\underline{\underline{\bf{\red{Solution:-}}}}}$

$\sf \: The \: numbers \: of \: logs \: in \: the \: bottom \: row \: next \: row \: row \: next \: to \\ \sf it \: and \: so \: on \: from \: the \: sequence$

$\: \: \: \: \: \: \: \: \: \: \: \sf \: 20, \: 19, \: 18, \: 17, \: ........$

$\therefore \: \: \: \: \: \sf \: a_2 - a_1 = 19 - 20 = - 1$

$\: \: \: \: \: \: \: \: \: \: \sf \: a_3 - a_2 = 18 - 19 = - 1$

$\: \: \: \: \: \: \: \: \: \: \sf \: a_4 - a_3 = 17 - 18 = - 1$

$\sf \: i.e., \: a_k+1 - a_k \: is \: he \: same \: everytime.$

$\sf \: So, \: the \: above \: sequences \: forms \: an \: AP.$

$\sf \: Here, \: \: a = 20$

$\: \: \: \: \: \: \: \: \sf \: d = - 1$

$\: \: \: \: \: \: \: \sf \: S_n = 200$

$\sf \: We \: know \: that \:$

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

$\implies \: \: \: \: \: \: \sf \: 200 = \frac{n}{2}[40 - n + 1] \\$

$\implies \: \: \: \: \: \: \: \: \sf200 = \frac{n}{2}(41 - n) \\$

$\implies \: \: \: \: \: \: \: \: \: \: \: \sf400 = n \: (41 - n)$

$\sf \: Cross \: - \: multiplying$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: 400 = 41n - n {}^{2}$

$\implies \: \: \: \: \: \: \: \: \: \: \: \sf \: n {}^{2} - 41n + 400 = 0 \\ \implies \: \: \: \: \: \sf \: n {}^{2} - 25n - 16n + 400 = 0 \\ \implies \: \: \: \sf \: n(n - 25) - 16(n - 25) = 0 \\ \implies \: \: \: \: \: \: \: \sf(n - 25)(n - 16) = 0 \\ \sf \implies \: \: \: \: \: n - 25 \: \: \: \: or \: \: \: \: \: \: \: n = 16$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: n \: = \: 25 \: 16$

$\sf \: Hence, \: the \: number \: of \: rows \: is \: either \: \red{ 25} \: or \: \red{16}.$

$\sf \: Now, \: Number \: of \: logs \: in \: top \: row$

$\sf \: \: \: \: \: \: = \: Number \: of \: logs \: in \: 25th \: row$

$\sf \: \: \: \: \: \: \: \: = a_25$

$\: \: \sf \: \: \: \: \: \: = \: a + (25 - 1)d$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: | ∵\sf \: a_n = a + (n - 1)d$

$\sf \: \: \: \: \: \: = \: a + 24d$

$\sf \: \: \: \: \: \: = \: 20 + 24 \: ( - 1)$

$\sf \: \: \: \: \: \: = \: 20 - 24$

$\: \: \: \: \: \: \sf = \: - \: 4$

$\sf \: which \: is \: not \: possible. \\ \: \: \: \: \: \sf \: Therefore, \: n = 16.$

$\: \sf \: and,$

$\: \: \: \sf \: Number \: of \: logs \: in \: top \: row$

$\sf \: \: \: \: \: \: \: = Number \: of \: logs \: in \: 16th \: row$

$\: \: \: \: \: \sf = a_16 \\ \sf \: \: \: \: \: = \: a + (16 - 1)d \\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: | ∵ a_n = a + (n - 1)d$