Sequences and series

Geometric sequence

A geometric sequence is a sequence of numbers in which the ratio between two consecutive terms is constant.

General term: $u_n=u_1\,r^{n-1}$

Common ratio: $r=\frac{\textstyle u_2}{\textstyle u_1}$

Find the common ratio $r$ and write down the general term of the following geometric sequence:

(a) $3,~ 6,~ 12,~ 24,~\ldots$

(b) $5,~ 15,~ 45,~ 135,~\ldots$

Solutions: (a) $r=2,~~ u_n=3\cdot 2^{n-1}$; (b) $r=3,~~ u_n=5\cdot 3^{n-1}$; (c) $r=\frac{1}{2},~~ u_n=80\cdot \left(\frac{1}{2}\right)^{n-1}$

Write down the general term and draw the graph of the following geometric sequence:

(a) $8,~ 12,~ 18,~ 27,~\ldots$

(b) $8,~ -4,~ 2,~ -1,~\ldots$

Solutions: (a) $u_n=8\cdot\left(\frac{3}{2}\right)^{n-1}$; (b) $u_n=8\cdot\left(-\frac{1}{2}\right)^{n-1}$

Write down the general term of the following geometric sequence:

(a) $-2,~ 2,~ -2,~ 2,~\ldots$

(b) $\frac{9}{2},~ \frac{3}{4},~ \frac{1}{8},~ \frac{1}{48},~\ldots$

Solutions: (a) $u_n=-2\cdot (-1)^{n-1}=2\cdot(-1)^n$; (b) $u_n=\frac{9}{2}\cdot\left(\frac{1}{6}\right)^{n-1}$; (c) $u_n=\sqrt{2}\cdot\big(\sqrt{2}\big)^{n-1}=\sqrt{2}^{~n}$

A geometric sequence has the first term $u_1=-3$ and the common ratio $r=4$. Write down the general term and the first five terms.
Solutions: General term: $u_n=-3\cdot4^{n-1}$; the first five terms: $-3,~ -12,~ -48,~ -192,~ -768$

A geometric sequence has the general term $u_n=6\cdot5^{n-1}$. Find the terms $u_4$ and $u_7$.
Solutions: $u_4=750,~~ u_7=93\,750$

A geometric sequence has the terms $5120,~ 2560,~ 1280,~ 640~\ldots$ Find the terms $u_7,~ u_{11}$ and $u_{15}$.
Solutions: $u_7=80,~~ u_{11}=5,~~ u_{15}=\frac{5}{16}$

A geometric sequence has the terms $u_1=7$ and $u_4=189$. Find the terms $u_5$ and $u_6$.
Solutions: $u_5=567,~~ u_6=1701$

In a geometric sequence the third term is 36 and the sixth term is 288. Find the first term and the common ratio.
Solutions: $u_1=9,~~ r=2$

The second term in a geometric sequence is 3 and the seventeenth term is $147\sqrt{7}$. Find the value of $u_{20}$.
Solution: $u_{20}=1029$

A geometric sequence has the first term $u_1=4$ and the common ratio $r=3$. Find out how many terms are less than 1000.
Solution: The first 6 terms are less than 1000.

A geometric sequence has the terms $\sqrt{3},~3,~ 3\sqrt{3},~9,~\ldots$ Write down the general term and find out how many terms are less than $10\,000$.
Solutions: $u_n=\sqrt{3}^{~n}$, the first 16 terms are less than $10\,000$.

A geometric sequence has the terms $u_1=2\sqrt{2}$ and $u_2=2$. Write down the general term and find out how many terms are greater than $10^{-6}$.
Solutions: $u_n=4\cdot\left(\frac{\sqrt{2}}{2}\right)^n$, the first 43 terms are greater than $10^{-6}$.

Insert four numbers between the given numbers 15 and 480 so that you obtain a finite geometric sequence.
Solutions: GS: $15,~ 30,~ 60,~ 120,~ 240,~ 480$

Insert three positive numbers between the given numbers 256 and 625 so that you obtain a finite geometric sequence.
Solutions: GS: $256,~ 320,~ 400,~ 500,~ 625$

Three numbers $a,~ b$ and $c$ form a finite geometric sequence if:

$b^2=a\,c$

Find the value of $x$ for which the following three numbers form a finite geometric sequence: $u_1=x-5,~~ u_2=x-1,~~ u_3=2x+4$
Solutions: (1) $x_1=7$, GS: $2,~ 6,~ 18$; (2) $x_2=-3$, GS: $-8,~ -4,~ -2$

Find the value of $x$ for which the following three numbers form a finite geometric sequence: $u_1=x-7,~~ u_2=x+3,~~ u_3=4x-3$
Solutions: (1) $x_1=12$, GS: $5,~ 15,~ 45$; (2) $x_2=\frac{1}{3}$, GS: $-\frac{20}{3},~ \frac{10}{3},~ -\frac{5}{3}$

Find the value of $x$ so that the following numbers are the first three terms of a geometric sequence: $u_1=\frac{\textstyle x-7}{\textstyle 9},~~ u_2=\frac{\textstyle x-4}{\textstyle 6},~~ u_3=\frac{\textstyle x}{\textstyle 4}$
Solutions: $x_1=16$, GS: $1,~ 2,~ 4$

Find the value of $x$ so that the following numbers are consecutive terms of a geometric sequence: $u_1=2,~~ u_2=2+\sqrt{x},~~ u_3=x-158$
Solutions: $x_1=400$, GS: $2,~ 22,~ 242$

Sum of the first n terms of a geometric sequence

The sum of the first $n$ terms of a geometric sequence can be calculated using the $s_n$ formula which has two forms:

$s_n=u_1\,\frac{\textstyle 1-r^n}{\textstyle 1-r}$ or

$s_n=u_1\,\frac{\textstyle r^n-1}{\textstyle r-1}$

You can use $\sum$ notation on your calculator to calculate the sum, too.

Find the sum of the first 6 terms of the geometric sequence:

(a) $u_n=8\cdot3^{n-1}$

(b) $u_n=5^n$

Solutions: (a) $S_6=2912$; (b) $S_6=19 530$

Find the sum of the first 7 terms of the geometric sequence:

(a) $5,~ 15,~ 45,~ 135,~\ldots$

(b) $1280,~ 640,~ 320,~ 160,~\ldots $

Solutions: (a) $S_7=5465$; (b) $S_7=2540$; (c) $S_7=2315$

Find the sum of the first 8 terms of the geometric sequence and write the result in exact form:

(a) $6,~ 2,~ \frac{2}{3},~ \frac{2}{9},~\ldots$

(b) $2,~ 2\sqrt{2},~ 4,~ 4\sqrt{2},~\ldots$

Solutions: (a) $S_8=\frac{6560}{729}$; (b) $S_8=30+30\sqrt{2}$

Calculate the sum:

(a) ${\displaystyle\sum_{r=1}^{15} 2^{r}}$

(b) ${\displaystyle\sum_{r=1}^{8} \left(\frac{3}{2}\right)^{r-1} }$

Solutions: (a) $S_{15}=65 534$; (b) $S_8=\frac{6305}{128}$

In a geometric sequence the first term is 3 and the seventh term is 15. Find the sum of the first 20 terms. Round the result to five significant digits.
Solution: $S_{20}=2074.5$

A geometric sequence has the general term $u_n=5\cdot 3^{n-1}$. Find the sum of all terms which are less than 100 000.
Solution: $S_{10}=147\,620$

A geometric sequence has the terms $u_1=25$ and $u_2=30$. Find the sum of all terms which are less than 100. Round your result to 4 significant figures.
Solution: $S_8=412.5$

The sum of the first 6 terms of a geometric sequence is 9555. The common ratio of this sequence is 4. Find the first term of this sequence.
Solution: $u_1=7$

A finite geometric sequence has the first term 4 and the second term 20. The sum of this sequence is 78 124. Find the number of terms in this sequence.
Solution: $n=7$

Compound interest

Adding interest to your bank deposit can be modelled using the geometric sequence. If the interest is compounded yearly, the value of your deposit after $n$ years will be:

$u_n=u_0\,\left(1+\frac{\textstyle p}{\textstyle 100}\right)^n$ or

$FV=PV\,\left(1+\frac{\textstyle p}{\textstyle 100}\right)^n$

If your calculator has the Finance Solver function, you can use it for all calculations in this section.

John Smith deposits 1000 € in his account at ABC-Bank. This bank compounds his deposit annually at 7% interest rate per annum (p.a.). How much will he have in his account:

(a) after 1 year,

(b) after 2 years,

Solutions: (a) 1070 €; (b) 1144,90 €; (c) 1402,55 €

Jane Doe deposits 14 790 € in The Cayman National Bank. The interest rate is 9% p.a. compounding annually. Find the final amount of her deposit after 8 years. How much will be the total interest?
Solutions: She will have 29 470 €. The total interest will be 14 680 €.

Joe Bloggs deposited 3292 £ in his savings account. After 10 years he found out that he has 4873 £ in his account. Calculate the annual interest rate.
Solution: The interest rate is 4% (compounded annually).

Uncle Sam deposited 15 800 $ in his bank account. The annual interest rate is 6% (compounded annually). In how many years will he have 33 700 $ in his account?
Solution: In 13 years.

John Bull invested 52 500 £ in stock with annual growth rate of 7.5%. In how many years will he have the amount of 87 100 £?
Solution: In 7 years.

The Best National Bank compounds deposits annually at 8% (p.a.). How many years would it take for your deposit to double? Round the number of years to nearest integer.
Solution: It'll take 9 years.

A bank compounds deposits annually at 7% (p.a.). How many years would it take for your deposit to increase by 50%? Round the number of years to nearest integer.
Solution: It'll take 6 years.

If the interest is compounded $k$ times in a year, the value of your deposit after $n$ years will be:

$FV=PV\,\left(1+\frac{\textstyle p}{\textstyle 100k}\right)^{kn}$

Finance Solver can solve this, too.

You deposited 12 000 € in your bank account. The interest rate is 6% (p.a.). How much will you have in your account after five years:

(a) if the interest is compounded yearly,

(b) if the interest is compounded monthly?

Solutions: (a) 16 058.71 €; (b) 16 186.20 €

Ivan Ivanovich Ivanov deposited 20 000 € in his account at The Swiss Royal Bank. The interest rate is 5% (p.a.). How much will he have in his account after ten years:

(a) if the interest is compounded yearly,

(b) if the interest is compounded half-yearly,

Solutions: (a) 32 577.89 €; (b) 32 772.33 €; (c) 32 872.39 €

Finance Solver

Financial calculations using TI-nspire

Financial calculations using Casio fx-GC50

Our friend wants to buy a car but she doesn't have enough money, so she is going to take out a loan of 15 000 €. The bank charges interest at a rate of 5% p.a. Calculate her monthly payment if she plans to repay the loan:

(a) in three years,

(b) in six yeaars.

Solutions: (a) 449.56 €; (b) 241.57 €

Annie wants to buy a small apartment. She is going to take out a loan of 50 000 € and she plans to repay it in 10 years. Her bank charges interest at a rate of 6% p.a.

(a) Calculate her monthly payment.

(b) Calculate the total amount of all her payments.

Benny wants to buy a small apartment, too. He is going to take out a loan of 50 000 € and he plans to repay it in 15 years. His bank charges interest at a rate of 5% p.a.

(d) Calculate the total amount of all his payments.

Solutions: (a) 555.10 €; (b) 66 612.30 €; (c) 395.40 €; (d) 71 171.43 €

Anastasia wanted to save some money. She decided to deposit 95 € in the beginning of each month for the next 5 years. Her bank has the interest rate of 2% p.a.

(a) Calculate the total amount she has in her bank after 5 years.

(b) Calculate the total amount of her deposits.

Boris decided to take out a loan of 6 000 €. He will repay it in 5 years in monthly payments. His bank charges interest at a rate of 2% p.a.

(d) Calculate his monthly payment.

(e) Calculate the total amount he will repay to his bank.

(f) Calculate the amount of interest he will have to pay.

Solutions: (a) 5 999.48 €; (b) 5 700 €; (c) 299.48 €; (d) 105.17 €; (e) 6 309.99 €; (f) 309.99 €