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…

(i) \(\displaystyle \mathrm{a}_{10}=1+9 \times 2=19\)

(ii) \(\displaystyle \mathrm{a}_{4}(\) from bottom \(\displaystyle )=19+3 \times(-2)=13\)

(iii) (a) \(\displaystyle \mathrm{S}_{8}=\frac{8}{2} \times[2 \times 1+7 \times 2]\) \[\text { = } 64 \]

(iii) (b) Number of triangles from \(\displaystyle 5^{\text {th }}\) row to \(\displaystyle 10^{\text {th }}\) row \(\displaystyle =\mathrm{S}_{10}-\mathrm{S}_{4}\) \[\begin{aligned} & =\frac{10}{2} \times[2 \times 1+9 \times 2]-\frac{4}{2} \times[2 \times 1+3 \times 2] \\ & =84 \end{aligned} \] \[\text { Number of triangles in first } \left.4 \text { rows, } \mathrm{S}_{4}=\frac{4}{2} \times[2 \times 1+3 \times 2] .\right] . \] \[\text { Required number of triangles }=84-16=68 \]