# Arithmetic and Geometric Sequences

Mathematics I · Arithmetic and Number Theory · https://tryals.app/learn/mathematics-i/arithmetic-and-geometric-sequences

## Arithmetic Sequences

A **sequence** is an ordered list $a_1, a_2, a_3, \dots$. An **arithmetic sequence** adds a fixed **common difference** $d$ each step.

$a_n = a_1 + (n-1)d$

For $2, 5, 8, 11, \dots$, the common difference is $d = 3$.

An **arithmetic series** is the sum of the first $n$ terms. The sum equals the number of terms times the average of the first and last:

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\big(2a_1 + (n-1)d\big)$

Young Gauss used this to add $1$ to $100$ instantly: $\frac{100}{2}(1 + 100) = 5050$.

## Geometric Sequences & Comparison

A **geometric sequence** multiplies by a fixed **common ratio** $r$ each step:

$a_n = a_1 r^{n-1}$

For $3, 6, 12, 24, \dots$, the ratio is $r = 2$. Its finite sum is:

$S_n = a_1 \frac{r^n - 1}{r - 1} \quad (r \neq 1)$

When $|r| < 1$, the infinite series converges to $\frac{a_1}{1 - r}$ ($1 + \frac{1}{2} + \frac{1}{4} + \dots = 2$).

| Feature | Arithmetic | Geometric |
| --- | --- | --- |
| **Growth** | Additive ($+d$) | Multiplicative ($\times r$) |
| **Graph** | Linear (straight line) | Exponential (curve) |

> **Common pitfall:** Mixing up formulas. Always identify whether terms **add** or **multiply** before applying a sum formula.

## Practice questions

6 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. A rumour spreads by each person telling two others; a savings account adds a fixed £50 a month. Why is the comparison never close in the long run, whatever the starting amounts?

A. Because the rumour's increment grows with the total and savings' does not
B. Because a doubling process necessarily begins from a bigger number than a fixed monthly sum
C. Because the savings account has an upper limit that the rumour does not
D. Because rumours spread faster than money accumulates for social reasons

**Answer:** A. Because the rumour's increment grows with the total and savings' does not

**Why:** One step is constant and the other is a proportion of a growing total, so the second compounds and the first does not. Start the savings at £1,000 a month and the doubling rumour still passes it inside a month, which is the arithmetic behind compound interest, epidemics and why O(2ⁿ) is hopeless.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/a-rumour-spreads-by-each-person-telling-two-others-a-savings-account

### 2. An arithmetic sequence adds a fixed amount each step; a geometric one multiplies by a fixed factor. Why does that difference make geometric growth eventually overtake any arithmetic sequence, however large its common difference?

A. Because a fixed step is eventually outpaced by one that keeps growing.
B. Because the two grow at the same rate, and the apparent difference is an artefact of how they are plotted.
C. Because arithmetic sequences are bounded above while geometric ones are not.
D. Because geometric sequences start from larger initial values, which gives them a permanent advantage.

**Answer:** A. Because a fixed step is eventually outpaced by one that keeps growing.

**Why:** The arithmetic sequence adds d at step 5 and at step 5000; the geometric one adds a proportion of its current value, so its increment grows exactly as fast as it does. Take 1000n against 2ⁿ: the arithmetic leads 10,000 to 1024 at n = 10 and trails 20,000 to a million at n = 20. Both are unbounded, the difference is the rate.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/an-arithmetic-sequence-adds-a-fixed-amount-each-step-a-geometric-one

### 3. Order the steps of finding the sum 3 + 6 + 12 + ... + 384.

**Answer:**

1. Identify the first term as 3 and the common ratio as 2
2. Solve 3 x 2 to the power n - 1 = 384 to find that there are 8 terms
3. Apply the finite geometric sum formula S_n = a_1 (r to the n, minus 1) divided by (r - 1)
4. Compute 3 x (256 - 1) divided by 1, giving 765

**Why:** Identify a_1 and r, count the terms, then apply the formula. Here 384 = 3 x 2 to the seventh, so n = 8, and S_8 = 3(2 to the eighth minus 1)/(2 - 1) = 3(255) = 765. Adding the eight terms by hand confirms it.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/order-the-steps-of-finding-the-sum-3-6-12-384

### 4. An arithmetic sequence has first term 4 and common difference 7. Find the twentieth term.

**Answer:** 137

**Why:** a_20 = 4 + 19(7) = 4 + 133 = 137. The factor is n - 1, not n, because the first term needs no steps at all, using 20 x 7 gives 144 and is the standard off-by-one error in this formula.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/an-arithmetic-sequence-has-first-term-4-and-common-difference-7-find

### 5. Find the sum of the first 30 terms of the arithmetic sequence 2, 5, 8, 11, ...

**Answer:** 1365

**Why:** a_30 = 2 + 29(3) = 89, so S_30 = (30/2)(2 + 89) = 15 x 91 = 1365. The formula says the sum is the number of terms times the average of the first and last. Gauss's pairing trick, and a much safer route than adding thirty numbers.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/find-the-sum-of-the-first-30-terms-of-the-arithmetic-sequence-2-5

### 6. A geometric sequence has first term 5 and common ratio 3. Find the sixth term.

**Answer:** 1215

**Why:** a_6 = 5 x 3 to the fifth = 5 x 243 = 1215. Again the exponent is n - 1: the first term has been multiplied by r zero times. Writing 5 x 3 to the sixth would give 3645, the same off-by-one error in multiplicative dress.

Page: https://tryals.app/practice/mathematics-i/arithmetic-and-geometric-sequences/a-geometric-sequence-has-first-term-5-and-common-ratio-3-find-the