Lesson 11 · Targeted Review

Filling the Gaps

Four weak spots — tackled directly.
This session is built from your actual results across lessons 7–10. Four clusters of problems gave you trouble: geometric sequences with fractional or decimal ratios, identifying sequence types from scratch, equations and inequalities with fractional coefficients, and finding the common difference from non-first terms. All four appear here.

Part A

Geometric sequences — when the ratio is a fraction or decimal
The method is identical to integer ratios — the only extra care is the arithmetic. Step 1: find $r$ by dividing any term by the one before it. Step 2: write $a_n = a_1 \cdot r^{n-1}$. Step 3: evaluate carefully, writing fractions as fractions.
  1. A sequence begins: 96, 48, 24, __, __. Find the two missing terms, write the formula for $a_n$, and find $a_7$.
  2. A sequence begins: 1 000, 600, 360, __, __. Find $r$, complete the sequence, write the formula for $a_n$, and find $a_6$.
  3. A sequence has $a_1 = 500$ and $r = \dfrac{4}{5}$. (a) Write the formula for $a_n$. (b) Find $a_4$. (c) Find the smallest $n$ for which $a_n < 100$. (Hint: calculate $a_5, a_6, \ldots$ one by one.)
  4. A horse supplement is given at 15 mL on day 1. Each day, the dose is $\dfrac{3}{4}$ of the previous day's dose. (a) Write the formula for the dose on day $n$. (b) On which day does the dose first fall below 2 mL?

Part B

Identify first — then solve
Run both tests before doing anything else.
Difference test: subtract consecutive terms — if the gap is always the same → arithmetic.
Ratio test: divide consecutive terms — if the result is always the same → geometric.
If neither test gives a constant → neither type.
  1. For each sequence below, run the difference test AND the ratio test, then label it: arithmetic, geometric, or neither. Do not calculate anything else yet.
    (a)   4,  12,  36,  108
    (b)   50,  41,  32,  23
    (c)   3,  5,  8,  12
  2. A pattern: 5, 10, 20, 40, 80. (a) Run both tests and state the type. (b) Write the formula for $a_n$. (c) Find $a_{10}$.
  3. Mia tracks her weekly riding practice: week 1: 20 min, week 2: 28 min, week 3: 36 min, week 4: 44 min. (a) Run both tests. Arithmetic or geometric? (b) Write the formula for $a_n$. (c) In which week does she first exceed 90 minutes?

Part C

Equations and inequalities with fractional and decimal coefficients
When the variable term is hidden in a subtraction — like $300 - 2.5k$ — rewrite it first: $-2.5k + 300$. That makes the negative coefficient visible before you solve. Then proceed as normal, flipping the inequality only when dividing or multiplying by a negative.
  1. Solve for $x$: $\quad \dfrac{3}{4}x + 1 = 10$
  2. Solve for $t$: $\quad 180 - 0.5t = 80$
  3. Solve the inequality and describe the solution in words: $\quad 240 - 2.5k > 165$
  4. Solve the inequality. Circle the step where you flip the sign and write one sentence explaining why. $$-\frac{n}{3} + 12 \leq 6$$

Part D

Arithmetic sequences — non-first terms and constraints
To find $d$ from two terms that are not $a_1$: count the number of steps between them (that's the difference in their positions), then divide the difference in values by the number of steps. If $a_3$ and $a_8$ are given, there are $8 - 3 = 5$ steps.
  1. A number pattern has its 2nd term equal to 11 and its 6th term equal to 27. Find $d$, find $a_1$, and write the formula for $a_n$.
  2. A pattern has $a_3 = 19$ and $a_8 = 44$. Find $d$, find $a_1$, and find $a_{15}$.
  3. A sequence has $a_1 = 60$ and $d = -7$. (a) Write the formula for $a_n$ in simplified form. (b) Find the first term that is negative, and state which position $n$ it is.
Show answers (with topic tags)
  1. geom. sequence $r = \tfrac{1}{2}$; missing terms: 12 and 6; $a_n = 96 \cdot \bigl(\tfrac{1}{2}\bigr)^{n-1}$; $a_7 = 96 \div 64 = \tfrac{3}{2} = 1.5$
  2. geom. sequence $r = 0.6$; missing terms: 216 and 129.6; $a_n = 1000 \cdot 0.6^{n-1}$; $a_6 = 1000 \cdot 0.6^5 = 1000 \cdot 0.07776 = 77.76$
  3. geom. sequence (a) $a_n = 500 \cdot \bigl(\tfrac{4}{5}\bigr)^{n-1}$ · (b) $a_4 = 500 \cdot \bigl(\tfrac{4}{5}\bigr)^3 = 500 \cdot \tfrac{64}{125} = 256$ · (c) $a_8 \approx 104.9$, $a_9 \approx 83.9$ → first below 100 is $a_9$, so $n = 9$
  4. geom. sequence (a) $a_n = 15 \cdot \bigl(\tfrac{3}{4}\bigr)^{n-1}$ · (b) $a_8 \approx 2.00$ mL (just at the boundary), $a_9 \approx 1.50$ mL → day 9 is the first day below 2 mL
  5. identification (a) Differences: 8, 24, 72 — not constant. Ratios: 3, 3, 3 — constant. → Geometric, $r = 3$ · (b) Differences: −9, −9, −9 — constant. → Arithmetic, $d = -9$ · (c) Differences: 2, 3, 4 — not constant. Ratios: $\tfrac{5}{3}$, $\tfrac{8}{5}$, $\tfrac{12}{8}$ — not constant. → Neither
  6. geom. sequence (a) Differences: 5, 10, 20, 40 — not constant. Ratios: 2, 2, 2, 2 — constant. → Geometric, $r = 2$ · (b) $a_n = 5 \cdot 2^{n-1}$ · (c) $a_{10} = 5 \cdot 2^9 = 5 \cdot 512 = 2\,560$
  7. arith. sequence (a) Differences: 8, 8, 8 — constant. → Arithmetic, $d = 8$ · (b) $a_n = 20 + (n-1) \cdot 8 = 12 + 8n$ · (c) $12 + 8n > 90 \Rightarrow 8n > 78 \Rightarrow n > 9.75 \Rightarrow$ week 10
  8. equation $\tfrac{3}{4}x = 9 \Rightarrow x = 9 \times \tfrac{4}{3} = 12$
  9. equation $-0.5t = -100 \Rightarrow t = 200$
  10. inequality Rewrite: $-2.5k > 165 - 240 = -75 \Rightarrow k < 30$ (divided by $-2.5$, flip). Solution: $k < 30$, i.e.\ $k$ is less than 30.
  11. inequality $-\tfrac{n}{3} \leq -6 \Rightarrow n \geq 18$ (multiply both sides by $-3$, flip because multiplying by a negative). Solution: $n \geq 18$.
  12. arith. sequence Steps from position 2 to position 6: $6 - 2 = 4$ steps. $d = (27 - 11) \div 4 = 4$. $a_1 = 11 - 4 = 7$. $a_n = 7 + (n-1) \cdot 4 = 3 + 4n$.
  13. arith. sequence Steps from position 3 to position 8: $8 - 3 = 5$ steps. $d = (44 - 19) \div 5 = 5$. $a_1 = 19 - 2 \times 5 = 9$. $a_{15} = 9 + 14 \times 5 = 79$.
  14. arith. sequence (a) $a_n = 60 + (n-1)(-7) = 67 - 7n$ · (b) Negative when $67 - 7n < 0 \Rightarrow n > \tfrac{67}{7} \approx 9.57 \Rightarrow n = 10$. First negative term: $a_{10} = 67 - 70 = -3$.
What's next → Lesson 12: Three Rich Problems
Three longer problems with no single obvious method — the kind that show you something new about how mathematics works. They pull together everything from lessons 1–11.
← All lessons
Focus 25:00

Ask your tutor