Arithmetic Progressions

State which of the following sequences are arithmetic progressions and, where applicable, give the common difference:

1. a) $4, 7, 10, 13, \ldots$

1. b) $2, 4, 8, 16, \ldots$

1. c) $20, 17, 14, 11, \ldots$

1. d) $1, 1, 1, 1, \ldots$

1. e) $3, 7, 12, 18, \ldots$

An arithmetic progression has first term $a_1 = 5$ and common difference $d = 3$.

1. a) Write its first five terms.

1. b) Write the general term $a_n$.

1. c) Calculate $a_{10}$.

In an arithmetic progression, $a_1 = 2$ and $a_5 = 14$.

1. a) Calculate the common difference.

1. b) Write the general term.

1. c) What position does the term $26$ occupy?

A savings account at $3 \%$ simple annual interest starts with an initial capital of $\$40\,000$ that is not withdrawn: each year the same interest on the initial capital is added.

1. a) Why do the capital amounts year after year form an arithmetic progression? State its first term and common difference.

1. b) Write the general term for the capital accumulated after $n$ years.

1. c) How much capital will there be after $6$ years? And after $15$ years?

The figure uses bars to represent the first six terms of an arithmetic progression.

1. a) Read the first term $a_1$ and the common difference $d$ from the graph, and explain why the heights of the bars lie on a straight line.

1. b) Write the general term $a_n$.

1. c) Calculate $a_{20}$ and the sum of the first $20$ terms.

Consider the arithmetic progression $3, 7, 11, \ldots$

1. a) Calculate the sum of its first $10$ terms.

1. b) Check the result by first calculating $a_{10}$ and applying the sum formula.

For the end-of-year trip, a student decides to save using a staircase system: $\$10$ in the first week and, each subsequent week, $\$5$ more than the previous one, as shown in the figure.

1. a) How much will she save in week $12$?

1. b) How much will she have accumulated in total after those $12$ weeks?

1. c) The trip costs $\$650$. Will saving for $16$ weeks be enough? Justify your answer.

Insert four arithmetic means between $3$ and $18$ (that is, construct an arithmetic progression of six terms whose extremes are $3$ and $18$).

1. a) Calculate the common difference of the progression.

1. b) Write the six terms.

1. c) Check that the sum of the six terms matches the formula $S_n = \frac{(a_1 + a_n) \times n}{2}$.

A park amphitheatre has $22$ seats in the first row, and each subsequent row has $4$ seats more than the previous one.

1. a) How many seats are in row $8$?

1. b) How many seats are there in total in the first $8$ rows?

1. c) A total of $600$ seats are needed for a concert. What is the minimum number of complete rows required?

Consider the arithmetic progressions $a_n$, with $a_1 = 7$ and common difference $2$, and $b_n$, with $b_1 = 40$ and common difference $-1$.

1. a) Write the general term for each one.

1. b) Is there any value of $n$ for which $a_n = b_n$? Calculate it.

1. c) Interpret the result: describe in words how each progression evolves and what the common term means.

How many terms of the arithmetic progression $5, 9, 13, \ldots$ must be added to obtain $377$?

1. a) Write the general term $a_n$ and the expression for the sum $S_n$ of the first $n$ terms.

1. b) Set up the equation $S_n = 377$ and solve it. Why must one of the solutions to the quadratic equation be discarded?

The lengths of the three sides of a right-angled triangle are in arithmetic progression with common difference $d > 0$.

1. a) Express the three sides in terms of the middle side $b$ and the common difference $d$, and apply Pythagoras' theorem.

1. b) Show that the sides are necessarily proportional to $3$, $4$ and $5$.

1. c) Give a concrete example of a right-angled triangle whose sides form an arithmetic progression with common difference $7$.

State with reasons whether the following statements about arithmetic progressions are true or false, proving them or giving a counterexample:

1. a) "Three numbers $a$, $b$ and $c$ are in arithmetic progression if and only if $b = \frac{a + c}{2}$."

1. b) "In an arithmetic progression, the sum of two terms equidistant from the extremes is always the same: $a_1 + a_n = a_2 + a_{n - 1} = a_3 + a_{n - 2} = \ldots$"

1. c) "If the same quantity is added to all the terms of an arithmetic progression, the common difference changes."

A father offers his daughter two monthly allowance plans:

• Plan A: $\$400$ in the first month, increasing by $\$40$ each month.

• Plan B: $\$160$ in the first month, increasing by $\$60$ each month.

1. a) Write the general term for the allowance under each plan in month $n$.

1. b) In which month does the monthly allowance of both plans coincide for the first time?

1. c) In which month does the cumulative total of plan B exceed that of plan A for the first time? Justify your answer using the sum formulas and reasoned trial and error.