1. Find the Rule: Are you adding ($d$) or multiplying ($r$)?
2. Identify $a_1$: This is the first number in the list.
3. The $(n-1)$ Rule: Remember that to get to the 5th term, you only take 4 jumps!
4. The Sigma Machine ($\sum$): The bottom number is where you start, the top is where you stop.
Find the missing terms and the $n$-th term.
The Sequence: $5, 12, 19, 26...$
A) What is the "Common Difference" ($d$)?
B) What is the 10th term ($a_{10}$)?
The Pruning: $100, 95, 90, 85...$
A) What is $d$?
B) Find $a_{20}$.
Use the formula $a_n = a_1 \cdot r^{(n-1)}$.
The Double: $3, 6, 12, 24...$
A) What is the "Common Ratio" ($r$)?
B) What is the 8th term?
The Reflection: $1000, 100, 10, 1...$
A) What is $r$?
B) What is the 6th term? (Note: It will be a fraction!).
In an Arithmetic sequence, if $d=0$, what happens to the sequence? In a Geometric sequence, if $r=1$, what happens? Why does this represent a "stagnant" heart?
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Expand and evaluate these sums.
$\sum_{i=1}^{4} (3i + 1)$
$\sum_{i=1}^{3} (2^i)$
A wall has 30 rows. The bottom row has 100 stones. Each row above it has 3 fewer stones.
1. How many stones are in the 30th row ($a_{30}$)?
2. Use the Arithmetic Sum formula $S_n = \frac{n}{2}(a_1 + a_n)$ to find the total number of stones.
Objective: Teach a younger student about sequences using their own steps.
The Activity:
1. Have them take 5 "Regular Steps" ($+1, +1, +1, +1, +1$).
2. Have them take 5 "Multiplication Jumps" (Jump 1 inch, then 2, then 4, then 8, then 16).
The Lesson: "Adding is for walking. Multiplying is for flying! God's grace helps us fly when we learn to multiply our love."
Response: ___________________________________________________________