a) To calculate the annual growth of the tree in the coming year:
Current growth:
\( 42 – 37 = 5 \, \text{m} \)

Next year’s growth (80% of current growth):
\( 5 \times 0.8 = 4 \, \text{m} \)

Thus:
The annual growth is \( 4 \, \text{m} \) [2]

b) To calculate the height of the tree 6 years from now:
Growth forms a geometric sequence with first term \( a = 4 \, \text{m} \) (from part a) and common ratio \( r = 0.8 \).
Sum of growth for 6 years:
\( S_6 = 4 \times \frac{1 – 0.8^6}{1 – 0.8} \)

Calculate:
\( 0.8^6 = 0.262144 \)
\( 1 – 0.262144 = 0.737856 \)
\( S_6 = 4 \times \frac{0.737856}{0.2} = 4 \times 3.68928 = 14.75712 \, \text{m} \)

Total height:
\( 42 + 14.75712 = 56.75712 \, \text{m} \)
To nearest cm:
\( 56.75712 \approx 56.76 \, \text{m} \) or \( 5676 \, \text{cm} \)

Alternative approach:
From last year (growth = 5 m):
\( S_7 = 5 \times \frac{1 – 0.8^7}{1 – 0.8} \)
\( 0.8^7 = 0.2097152 \)
\( S_7 = 5 \times \frac{1 – 0.2097152}{0.2} = 5 \times 3.951424 = 19.75712 \, \text{m} \)
Height = \( 37 + 19.75712 = 56.75712 \, \text{m} \approx 56.76 \, \text{m} \)

Thus:
The height is \( 56.76 \, \text{m} \) or \( 5676 \, \text{cm} \) [4]

c) To find the smallest possible value of \( k \):
Growth forms an infinite geometric series with first term \( a = 4 \, \text{m} \) and common ratio \( r = 0.8 \).
Sum of infinite series:
\( S_\infty = \frac{4}{1 – 0.8} = \frac{4}{0.2} = 20 \, \text{m} \)

Total height:
\( 42 + 20 = 62 \, \text{m} \)

Alternative approach:
From last year (growth = 5 m):
\( S_\infty = \frac{5}{1 – 0.8} = \frac{5}{0.2} = 25 \, \text{m} \)
Height = \( 37 + 25 = 62 \, \text{m} \)

Thus:
The smallest possible value of \( k \) is \( 62 \, \text{m} \) [3]