In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on.
Based on given information, answer the following questions using Arithmetic Progression.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding stimulus
Model Answer
The number of triangles in each row forms an AP: 1, 3, 5, 7, … with first term $a = 1$ and common difference $d = 2$.
(i) The bottom-most row is the 10th row.
$a_{10} = 1 + (10-1) \times 2 = 1 + 18 = \mathbf{19}$
There are 19 triangles in the bottom-most row.
(ii) The 4th row from the bottom is the 7th row (from top).
$a_7 = 1 + (7-1) \times 2 = 1 + 12 = \mathbf{13}$
There are 13 triangles in the 4th row from the bottom.
(iii) Total triangles up to 8th row:
$$S_8 = \frac{8}{2}[2(1) + (8-1) \times 2] = 4[2 + 14] = 4 \times 16 = \mathbf{64}$$
Total number of triangles till 8th row = 64.
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Explanation
- The rows from top form an AP with $a=1$, $d=2$, so the $n$th row has $(2n-1)$ triangles.
- For (ii), "4th from bottom" in a 10-row triangle = row number $10 - 4 + 1 = 7$ from top.
- For (iii), use the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$. Examiners expect all steps shown clearly.