I was given:
$$\mathring e_x = 12.38763065, \quad \mathring e_{x+3} = 11.65757292$$
and
$${}_2 q_{x+1} = 0.1201466209, \quad q_x = 0.05917676022$$
and I was asked to find:
$$\require{enclose} \mathring e_{x : \enclose{actuarial}{3}}$$
I want to use the formula:
$$\mathring e_x = \mathring e_{x : \enclose{actuarial}{n}} + {}_n p_x \; \mathring e_{x+n}$$
But I'm not sure if mathematically I'm allowed to use it. For example, for ${}_n p_x$ in the formula, I want to do $(3)(1-0.05917676022)$ but I'm not sure if that's allowed.
thanks!!
Your strategy is correct: the formula you cite, $$\require{enclose} \mathring e_x = \mathring e_{x : \enclose{actuarial}{n}} + {}_n p_x \; \mathring e_{x + n} \tag{1}$$ with the choice $n = 3$, yields $$\mathring e_{x : \enclose{actuarial}{3}} = \mathring e_x - {}_3 p_x \; \mathring e_{x+3}. \tag{2}$$ Since the only quantity that we are not given is ${}_3 p_x$, we have to compute it. Recall that $${}_n p_x = 1 - {}_n q_x, \tag{3}$$ that is to say, the probability that $(x)$ survives $n$ years is equal to the complement of the probability that $(x)$ dies within $n$ years. Since $q_x$ is the probability that $(x)$ dies within the next year, and ${}_2 q_{x+1}$ is the probability that $(x+1)$ dies within the next two years, the probability that $(x)$ dies within the next $3$ years is $$\begin{align} {}_3 q_x &= q_x + (p_x)({}_2 q_{x+1}) \\ &= q_x + (1 - q_x)({}_2 q_{x+1}). \tag{4} \end{align}$$ This is because either $(x)$ dies in the next year, or $(x)$ survives the first year to reach $(x+1)$ but dies within the next two years. Putting $(3)$ together with $(4)$ then gives us $$\begin{align} {}_3 p_x &= 1 - q_x - (1 - q_x)({}_2 q_{x+1}) \\ &= (1 - q_x)(1 - {}_2 q_{x+1}). \tag{5} \end{align}$$
We could also have obtained the identity $(5)$ by observing that $${}_3 p_x = (p_x)({}_2 p_{x+1}), \tag{6}$$ since $(x)$ survives $3$ years if they survive the first year to reach $(x+1)$, then survives an additional $2$ years. Then we apply $(3)$ to each factor on the right hand side, and we again get $(5)$.
All that remains is to substitute $(5)$ into $(2)$ and use the given information to compute the result, which I leave to you.