These 24 problems practice arithmetic and geometric sequences with explicit and recursive formulas (F-BF.A.2, F-IF.A.3), then sigma notation and finite series sums (A-SSE.B.4). Early problems are short and direct, and each one has a full worked answer.
The last six are word problems about savings, seating, a bouncing ball, raises and loans. Show every step, and check each series sum by adding a few terms by hand.
Problem 1: Arithmetic or geometric
Decide whether each sequence is arithmetic or geometric. Give the common difference or ratio.
- \(5,\,9,\,13,\,17\)
- \(3,\,6,\,12,\,24\)
- \(20,\,17,\,14,\,11\)
- \(81,\,27,\,9,\,3\)
Problem 2: Reading a graph
The graph shows the first five terms of a sequence.
- Decide whether the sequence is arithmetic or geometric.
- Write an explicit formula for \(a_{n}\).
- Find \(a_{12}\).
Problem 3: Explicit arithmetic rule
An arithmetic sequence has \(a_{1}=7\) and common difference \(d=5\). Write an explicit formula for \(a_{n}\) and find \(a_{20}\).
Problem 4: Explicit geometric rule
A geometric sequence has \(a_{1}=3\) and common ratio \(r=2\). Write an explicit formula for \(a_{n}\) and find \(a_{10}\).
Problem 5: Terms from a recursive rule
Write the first five terms of each sequence.
- \(a_{1}=4,\quad a_{n}=-2\cdot a_{n-1}\)
- \(b_{1}=50,\quad b_{n}=b_{n-1}-6\)
Problem 6: Writing a recursive rule
Write a recursive formula for each sequence.
- \(12,\,19,\,26,\,33\)
- \(80,\,40,\,20,\,10\)
Problem 7: How many terms
An arithmetic sequence starts \(8,\,13,\,18\) and continues with common difference 5 until its last term, 163. How many terms does it have?
Problem 8: Rule from two terms
An arithmetic sequence has \(a_{2}=9\) and \(a_{7}=39\). Find the common difference, \(a_{1}\), and an explicit formula.
Problem 9: Geometric from two terms
A geometric sequence of positive numbers has \(a_{2}=12\) and \(a_{5}=324\). Find \(r\), \(a_{1}\), and \(a_{8}\).
Problem 10: Evaluating sigma notation
Evaluate \(\sum_{k=1}^{6}(2k+1)\).
Problem 11: Geometric sigma
Evaluate \(\sum_{k=1}^{4}2\cdot 3^{k-1}\).
Problem 12: Writing sigma notation
Write each series in sigma notation.
- \(6+10+14+18+22\)
- \(3+6+12+24+48+96\)
Problem 13: Arithmetic series sum
Find the sum of the arithmetic series that starts \(4+9+14\) and continues with common difference 5 until the last term, 99.
Problem 14: Geometric series sum
Find the sum of the first 8 terms of the geometric series \(5+10+20+40\) that continues with ratio 2.
Problem 15: Ratio less than 1
Find the sum of the first 6 terms of the geometric series \(81+27+9+3\) that continues with ratio \(\frac{1}{3}\).
Problem 16: Finding the number of terms
An arithmetic series has \(a_{1}=2\) and \(d=3\). How many terms must be added to reach a sum of 155?
Problem 17: Weekly savings
Maya saves $5 in week 1. Each week after that she saves $3 more than the week before.
- How much does she save in week 12?
- How much has she saved in total after 12 weeks?
Problem 18: Theater seating
A school theater has 18 rows. Row 1 has 14 seats and each row has 2 more seats than the row in front of it.
- How many seats are in row 18?
- How many seats are in the theater?
Problem 19: Bouncing ball
Jamal drops a ball from a height of 10 feet. After each bounce, the ball rises to 60% of the height of the previous bounce.
- Write an explicit formula for the height of bounce \(n\).
- Find the height of the fifth bounce.
- Find the sum of the first five bounce heights.
Problem 20: Chain email
Noah emails a joke to 3 friends on day 1. On each following day, every person who got the joke the day before forwards it to 3 new people.
- How many people receive the joke on day 7?
- How many people have received it in total by the end of day 7?
Problem 21: Raises over ten years
Aisha starts a job at $42,000 per year and receives a 3% raise at the start of each year after the first. Round to the nearest cent.
- Find her salary in year 10.
- Find her total earnings over the first 10 years.
Problem 22: Savings account
Priya deposits $200 at the end of every month into an account that earns 0.5% interest per month. Find the balance right after her 12th deposit.
Problem 23: Car loan
Diego pays $300 per month for 36 months to repay a car loan at 0.5% interest per month. The amount borrowed equals the value today of all 36 payments, where a payment made \(k\) months from now is worth \(300(1.005)^{-k}\) today.
- Find the amount he borrowed.
- Find the total interest he pays.
Problem 24: Mortgage payment
Mei takes out a $150,000 mortgage for 15 years at 6% annual interest, which is 0.5% per month. The loan amount equals the value today of all 180 equal monthly payments \(M\).
- Write the loan amount as a geometric series sum in terms of \(M\), then find \(M\) to the nearest cent.
- Find the total amount paid over 15 years and the total interest.
See every chapter on the Algebra 2 math page.