![]() The interest received over 5 years is thus: We know that James’ principal sum is £2,000 and we found in part a) that his total balance after 5 years is £2,400. Total balance = £ (2,000 + (5×80)) = £ 2,400ī) The interest that James receives over 5years is equal to the difference between his total balance after 5years and his principal sum. We want to calculate his total balance after 5 years. Solution: a) As James withdraws the interest earned each year, his total balance forms an arithmetic sequence with first term a=2,000 and common difference d = 0.04×2000 = 80. The interest that he receives after 55 years.He withdraws and spends the interest earned every time it is paid. Using the above formula for the nth term of an arithmetic progression, the total balance at the end of n years is a + nd.Įxample: James deposits £2,000 into a bank which pays an annual interest rate of 4%. In general, for an interest rate of r per annum and principal balance of A, the total balance will form an arithmetic progression with first term A and common difference d=r × a. We can see that the total balance-the principal balance and the interest earned each year up to the present year-forms an arithmetic progression with first term a and common difference 0.15a. At the end of the second year you will therefore have a total of:Īt the end of the third year you will have:Īnd so on till n years i.e., (a(1+n×0.15)) This means that the interest paid each year is only ever paid on the principal balance. With simple interest, the key assumption is that you withdraw the interest from the bank as soon as it is paid and deposit it into a separate bank account. ![]() At the end of the first year you will have a total of: You are paid 15% interest on your deposit at the end of each year (per annum). Suppose you deposit an amount of a into a bank. The form of an arithmetic progression is a, a+d, a+2d, a+3d, a+4d so using these values of a and d the first five terms are: Use the formula of the arithmetic sequence.Įxample 2: Write down the first five terms of the arithmetic progression with first term 8 and common difference 7. Write the formula that describes this sequence. ![]() The general (nth) term of an arithmetic sequence, a n, with first term a 1 and common difference d, may be expressed explicitly as:Įxample 1: f a sequence has a first term of a 1 = 12 and a common difference d = −7. Sequence is defined as, F 0 = 0 and F 1 = 1 and F n = F n-1 + F n-2 ARITHMETIC SEQUENCESĪ sequence a1, a2, a3, …., an is an arithmetic sequence if there is a constant d for whichįor all integers n > 1, d is called the common difference of the sequence, and d = a n – a n-1 for all integers n > 1. ![]() Fibonacci Numbersįibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Harmonic SequencesĪ series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence. Where we’ll be studying arithmetic and geometric in detail. Note: The series is finite or infinite depending if the sequence is finite or infinite. Series a number of events, objects, or people of a similar or related kind coming one after another. is a sequence, then the corresponding series is given by either finite sequence or infinite sequence.
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