Arithmetic and Geometric sequence and series

Definitions A sequence is a list of numbers which follow a definite pattern or rule. If the rule is that each term, after the first, is obtained by adding a constant, d, to the previous term, then the sequence is called an arithmetic sequence, such as 2, 6, 10, 14, 18, 22, . . . , where d — 4. d is known as the common difference. If the rule is that each term, after the first, is obtained by multiplying the previous term by a constant, r, then the sequence is called a geometric sequence, such as, 4, 16, 64, 256, 1024, where r = 4. r is known as the common ratio. A series is the sum of the terms of a sequence. A series is finite if it is the sum of a finite number of terms of a sequence. A series is infinite if it is the sum of an infinite number of terms of a sequence.

EXAMPLE An arithmetic series is the sum of the terms of an arithmetic sequence, such as 2, 6, 10, 14, 18, 22, . . . , with d= 4 Find the sum of the first 15 terms of the series: 20+18+16+14 + Solution This is an arithmetic series since the difference d = -2. Therefore, a = 20, d = –2, n = 15, and hence, using formula S15=15/2[2(20) + (15-l)(-2)] = 90

Exercises Simple and Compound Interest 1. Suppose £5000 is invested for five years. Calculate the amount accumulated at the end of five years if interest is compounded annually at a nominal rate of (a) 5%, (b) 7%, (c) 10%. 2. A savings account of £10000 earns simple interest at 5% per annum. Calculate the value of the account (future value) after six years. 3. £2500 is invested at a nominal rate of interest of 5% per annum. Calculate the amount accumulated at the end of (a) 1 year, (b) 4.5 years, (c) 10 years, (d) 20 years. 4. Calculate the present value of £6000 that is expected to be received in three years' time with simple interest of 7.5% per annum. 5. How much is a sum £3500 worth at the end of five years if deposited at (a) 11 % simple interest and (b) 11 % compound interest, each calculated annually?