To solve this problem, let's break down the given information and calculate $\frac{1}{q_{[80]+1}}$.

Step 1: Understanding the Data

The table provides the number of lives at each age in a select and ultimate mortality table. The columns represent:

• $l_{[x]}$: Number of lives at the beginning of the age $x$.
• $l_{[x]+1}$: Number of lives after one year from age $x$.
• $l_{x+2}$: Number of lives two years after age $x$.

From the table:

• For $x = 80$:
  • $l_{[80]} = 1000$
  • $l_{[80]+1} = 950$
  • $l_{82} = 900$

Step 2: Mortality Rate Calculation

The problem states that $q_{[80]+1} = q_{[81]+1}$, and we need to calculate $\frac{1}{q_{[80]+1}}$.

Calculate $q_{[80]+1}$:

The mortality rate $q_{[80]+1}$ is given by the probability of death between age 81 and 82 for a life aged 81:

[ q_{[80]+1