The correct answer is A. 7 years.
The sum-of-the-years’-digits (SYD) method is a depreciation method that allocates a greater depreciation expense in the early years of an asset’s life and a smaller depreciation expense in the later years. The formula for SYD depreciation is:
$D_n = \dfrac{n(n+1)}{2}$
where $n$ is the number of years of the asset’s life.
In this case, the company wants the annual depreciation cost to be no more than 20% of the first cost. This means that the depreciation expense for each year must be less than or equal to 0.2($C_0$), where $C_0$ is the first cost of the asset.
We can use the SYD formula to solve for $n$, the number of years of the asset’s life, such that $D_n \leq 0.2($C_0$):
$0.2($C_0$) \leq \dfrac{n(n+1)}{2}$
$C_0 \leq \dfrac{n(n+1)}{4}$
$4C_0 \leq n(n+1)$
$4C_0 – n^2 – n + 0 \leq 0$
$(n-7)(n-4) \leq 0$
$n \geq 7$ or $n \leq 4$
Since the asset cannot have a negative life, the only possible value of $n$ is 7.
Therefore, the length of service life necessary if the depreciation used is the SYD method is 7 years.