So You Think You Can Add: Arithmetic Sequences
State the formula for the n-th term of an arithmetic sequence.
an=a1+(n−1)d
State the formula for the n-th term of a geometric sequence.
an=a1⋅rn−1
What does a recursive sequence require that an explicit sequence does not?
Recursive rules require the previous term. Explicit rules do not.
Describe how the graph of an arithmetic sequence differs from a geometric one.
Arithmetic → points form a straight line pattern.
Geometric → points form a curved exponential pattern.
You save $20 each week. Is this an arithmetic or geometric situation? Why?
What is arithmetic?
Reason: You add a constant amount each week.
Is the sequence 5, 9, 13, 17 arithmetic? If yes, give the common difference.
What is four?
d=9−5=4
Is 3, 6, 12, 24 geometric? If yes, give r.
What is 2? r=6/3=2
Identify whether the rule below is explicit or recursive:
an = an-1 + 4, a1 = 7
What is recursive?
If a graph of a sequence shows points decreasing in a straight-line pattern, what type of sequence is it?
What is arithmetic?
A bacteria culture triples each hour. Write the explicit rule for the population after n hours.
an=a1*3n−1
Find the 15th term of a sequence where a₁ = –2 and d = 6.
What is 82?
a15=−2+(15−1)(6)=−2+84=82
Find a₁ if a₄ = 81 and r = 3.
What is 3?
a4=81=a1(33)=27a1⇒ a1=3
Convert the explicit formula an= 12 – 3n into a recursive formula.
an=an−1−3, where a1=9
Graph the first five terms of the sequence an = 5(2n). Determine if the graph grows linearly or exponentially.
What is exponential growth?
Identify whether depreciation of a car by 15% per year is arithmetic or geometric and explain your reasoning.
What is geometric?
Reason: Depreciation is a percent change, meaning multiplication by a constant ratio < 1.
Two arithmetic sequences have the same 1st and 20th terms, but different differences. Explain how this is possible.
What are different slopes?
They have the same endpoint with different starting values.
Compare the growth of a geometric sequence with r = 1.02 and r = 5. Which grows more quickly and why?
The sequence with r=5 grows exponentially much faster than the sequence with r=1.02 because the multiplier each term is much larger.
Explain why recursive formulas may be inefficient for finding very large n.
Recursive formulas require computing all earlier terms to reach large n. Explicit formulas compute directly.
A graph appears exponential but begins with negative terms. Explain how this is possible for a sequence.
If the first term is negative, the early exponential values can also be negative (e.g., multiplying a negative a1 by a positive ratio r>1).
A job offers a salary increase of $1500 per year plus a 5% raise. Explain why this cannot be modeled by a pure arithmetic or geometric sequence.
It is a hybrid model because it combines:
a constant difference (+1500, arithmetic),
a constant percent increase (5%, geometric).
Decide whether the following student statement is true:
“Every linear function is an arithmetic sequence.” Justify your reasoning.
What is false?
Every arithmetic sequence is linear, but a function has infinite domain; a sequence has discrete domain (natural numbers only). Linear ≠ arithmetic
A student claims: “Any sequence that grows really fast must be geometric.” Evaluate this claim.
What is false?
A sequence may grow quickly for other reasons (e.g., quadratic, factorial). Not all fast growth is geometric.
Given two equivalent rules (one explicit, one recursive), argue which is more useful in real-world modeling and why.
Explicit is more efficient when dealing with large values and stepwise/dependent processes like finance; recursive is conceptually useful.
Decide whether the following graph could represent a geometric sequence. Justify your answer. (Provide a sketch.)
Could be geometric only if the ratio between successive y-values is constant.
If the graph does not show constant multiplicative jumps, then the answer is No.
A student claims that compound interest is “just a fancy arithmetic sequence.” Evaluate this statement.
What is false?
Compound interest is geometric, because each term is multiplied by a common ratio 1+r