The formula for a compound growth or decay is given by
A = A₀ × (1+growth rate)ⁿ
Where:
A = The final value after 'n' years
A₀ = The initial value
Growth rate = 5% = 0.05
We know that final value = 3 × Initial value
So that's A = 3A₀
Substitute this expression into the formula, we have
3A₀ = A₀ (1+0.05)ⁿ
3A₀ ÷ A₀ = (1.05)ⁿ
3 = (1.05)ⁿ
from here we can use the trial and error method:
take n = 1, then (1.05)¹ = 1.05
take n = 2, then (1.05)² = 1.1025
take n = 3, then (1.05)³ = 1.16
take n = 5, then (1.05)⁵ = 1.3
take n = 10, then (1.05)¹⁰ = 1.6
take n = 20, then (1.05)²⁰ = 2.7
take n = 30, then (1.05)³⁰ = 4.3 ⇒ too big
take n = 25, then (1.05)²⁵ = 3.4
take n = 24, then (1.05)²⁴ = 3.2
take n = 23, then (1.05)²³ = 3.1
take n = 22, then (1.05)²² = 2.9
so, the number of years it takes for the account to grow three times the initial value after 23 years.