Estimation, Rounding, and Sequences

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_n = \frac{1}{\sqrt{n} + \sqrt{n+2}}$ for all positive integers $n$. If $S = \sum_{n=1}^{98} a_n$, which of the following is closest to the value of $S$ when rounded to the nearest tenth?

An index value $I_0 = 1{}000$ increases over a 6-month period. For each month $n$ from 1 to 6, the index value $I_n$ is calculated by increasing the previous month's value $I_{n-1}$ by $5\%$ and then rounding the result to the nearest integer (with half-integers rounded up). Let $U_6 = 1{}000 \times (1.05)^6$ represent the exact unrounded compounded value at month 6.

Which of the following statements must be true? Select all such statements.

A financial forecasting model predicts monthly sales revenue using a sequence where the initial revenue in month 1 is $R_1 = \$80{}000$, and the revenue for each subsequent month $n$ (for $n \ge 2$) is calculated using the recursive formula $R_n = 0.75 R_{n-1} + 5{}000$. If each monthly revenue figure $R_n$ is rounded to the nearest hundred dollars before summing, what is the estimated total revenue, in dollars, for the first 4 months combined?

A sequence $a_1, a_2, a_3, \dots, a_n$ is defined by $a_1 = 1$ and $a_n = 2a_{n-1} + 3$ for all integers $n \geq 2$. What is the value of $a_4$?

A sequence of numbers begins with $t_1 = 4$. For all integers $n \geq 2$, each term is defined by $t_n = 3t_{n-1} - 2$. What is the value of $t_5$?

A sequence $a_1, a_2, a_3, \dots$ is defined by $a_1 = 3$ and $a_{n+1} = \sqrt{a_n + 12}$ for all integers $n \geq 1$. If $S_{100} = \sum_{k=1}^{100} a_k$, what is the value of $S_{100}$ rounded to the nearest integer?

A sequence of positive real numbers $a_1, a_2, a_3, \dots$ is defined by $a_1 = 3$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all integers $n \ge 1$. Which of the following statements must be true? Select all that apply.

A sequence of 50 numerical measurements $x_1, x_2, \dots, x_{50}$ is collected. Each measurement $x_i$ is rounded to the nearest tenth to produce a rounded value $r_i$. The sum of the 50 rounded values, $\sum_{i=1}^{50} r_i$, is equal to $250.0$. If $S$ represents the true sum of the unrounded measurements $\sum_{i=1}^{50} x_i$, what is the maximum possible percent error of the rounded sum relative to the true sum $S$, rounded to the nearest hundredth of a percent?

A sequence of 40 numbers $a_1, a_2, \dots, a_{40}$ is defined by the formula $a_k = 15 + \frac{3k}{8}$ for each integer $k$ from 1 to 40. Each term $a_k$ is rounded to the nearest integer to form a new sequence $b_1, b_2, \dots, b_{40}$. (Note: numbers ending in $.5$ are rounded up to the next integer.) What is the value of $\sum_{k=1}^{40} b_k - \sum_{k=1}^{40} a_k$?

A sequence of 24 positive numbers $a_1, a_2, \dots, a_{24}$ is defined by $a_n = \frac{120}{n(n+1)}$ for each integer $n$ from $1$ to $24$. Let $T = \sum_{n=1}^{24} a_n$ be the exact sum of all 24 terms. Which of the following statements must be true? Select all that apply.

A sequence of numbers $a_1, a_2, a_3, \dots$ is defined by $a_1 = 5$ and $a_{n+1} = a_n^2 - 2$ for all positive integers $n \ge 1$. Which of the following is the value of the sum $\sum_{n=1}^{4} \frac{1}{a_n}$, rounded to the nearest hundredth?

A sequence of numbers $a_1, a_2, a_3, \dots$ is defined by $a_1 = 50$ and $a_{n+1} = \frac{1}{3} a_n + 15$ for all positive integers $n \ge 1$. Each term $a_n$ is rounded to the nearest integer to produce a secondary sequence $b_n$. What is the value of $\sum_{n=1}^{4} b_n$?