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A savings account at $3 \%$ simple annual interest starts with an initial capital of $2.000\ \text{ €}$ that is not withdrawn: each year the same interest on the initial capital is added.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?

For the end-of-year trip, a student decides to save using a staircase system: $10\ \text{ €}$ in the first week and, each subsequent week, $5\ \text{ €}$ 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 $650\ \text{ €}$. 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 $S_n = \frac{(a_1 + a_n) \cdot n}{2}$.

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?

A father offers his daughter two monthly allowance plans:

• Plan A: $20\ \text{ €}$ in the first month, increasing by $2\ \text{ €}$ each month.

• Plan B: $8\ \text{ €}$ in the first month, increasing by $3\ \text{ €}$ 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.