Computing terms yields 1, 2, 4, 8, 16, 32, 64, 128. Since Pn – Pn-1 = Pn-1, the simpler recursive rule is Pn = 2Pn-1 for n >= 3. The explicit formula is Pn = 2ⁿ⁻¹ for n >= 2 ...
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With a = 2 and r = 4, explicit formula is tn = 2 x 4ⁿ⁻¹ and recursive formula is t1 = 2, tn = 4tn-1 for n >= 2. Solving 2 x 4ⁿ⁻¹ = 131072 gives 4ⁿ⁻¹ = 65536 ...
The sum of k consecutive natural numbers is k(2a + k – 1) / 2 = 100, meaning k(2a + k – 1) = 200. Testing odd factors gives only k = 5, yielding 18 + 19 + 20 + ...
The bounce heights form a GP with first term 48 metres and common ratio 0.6. Therefore, the fifth bounce height is 48 multiplied by 0.6 raised to 4, which equals 6.2208 metres. Hence, the required height is 6.2208 metres.