Answer:
Geometric sequences are used in several branches of applied mathematics to engineering, sciences, computer sciences, biology, finance...
Problems and exercises involving geometric sequences, along with detailed solutions and answers, are presented.
REVIEW OF GEOMETRIC SEQUENCES
2 , 8 , 32 , 128 , ...has been obtained starting from 2 and multiplying each term by 4. 2 is the first term of the sequence and 4 is the common ratio. \( 8 = 2 \times 4 \\ 32 = 8 \times 4 \\ 128 = 32 \times 4 \\ \text{and so on} \)
The terms in the sequence may also be written as follows
\( a_1 = 2 \\ a_2 = a_1 \times 4 = 2 \times 4 \\ a_3 = a_2 \times 4 = 2 \times 4^2 \\ a_4 = a_3 \times 4 = 2 \times 4^3 \\ \)
The n th term may now be written as \[ a_n = a_1 r^{n-1} \]
where a 1 is the first term of the sequence and r is the common ratio which is equal to 4 in the above example.
The sum of the first n terms of a geometric sequence is given by \[ s_n = a_1 + a_2 + a_3 + ... a_n = a_1 \dfrac{1 - r^n}{1-r} \] The sum S of an infinite (n approaches infinity) geometric sequence and when |r| < 1 is given by
\[ S = \dfrac{a_1}{1-r} \]
Arithmetic Series Online Calculator . An online calculator to calculate the sum of the terms in an arithmetic sequence.
