03 Series Today s Objectives Definition of

03 Series

Today’s Objectives • Definition of an arithmetic progression and the nth term • Sum of an arithmetic progression

Arithmetic Progressions (APs) • An arithmetic progression is when the difference between each term is a fixed real amount. • This difference is known as the common difference and we use the letter d. • Two examples of arithmetic progressions are; • 6, 10, 14, 18, … (Tn = 6 + (n – 1)4) • and – 2, – 6, – 10, – 14, … (Tn = – 2 + (n – 1)(– 4))

Arithmetic Progressions (APs) • So the general formula for the nth term of an arithmetic progression is: Tn = a + (n – 1)d where a is the first term and d is the common difference.

AP Example 1 • The 8 th term of an AP is 11 and the 15 th term is 21. What is the common difference and the nth term? • To answer this we will use the formula for the nth term, Tn = a + (n – 1)d, to find the common difference d and the first term a.

AP Example 1 • Solution:

AP Example 2 • The nth term of an AP is 12 – 4 n. What is the 1 st term and the common difference? • To find the 1 st term we will use the nth term when n = 1. To find the common difference we will find the 2 nd term when n = 2 and subtract the 1 st term from the 2 nd term (since T 1 + d = T 2).

AP Example 2 • Solution:

AP Example 3 • An AP with all positive terms, is such that T 1 T 10 = 12 and T 1/T 10 = 1/3. i) Find the common difference, ii) Find the value of the next term after the 10 th term which is a whole number. • i) To find the common difference we will use the nth term , Tn = a + (n – 1)d, to form simultaneous equations and solve for a and d. • ii) We must choose a value of n (n>10) to resolve any fractions in Tn.

AP Example 3 • i) Solution: All terms are positive so d > 0.

AP Example 3 • ii) Solution: From part i) above: Tn is a whole number. So, n – 1 is a multiple of 9.

Sum of an AP • Considering the sum of the first n terms of the sequence (i. e. the series), where a is the first term, d is the common difference and l is the last term: • But this is rather long.

Sum of an AP • So now consider the same expression backwards and add the two together: • Considering the RHS there are n terms. So we can simplify the RHS:

Sum of an AP • Simplified we have: • Hence the sum of the n terms of an arithmetic progression is given by:

Sum of an AP • Also, since the last term, l, can be written as: a + (n – 1)d, we can replace l in the above formula to get an alternative formula for the sum of n terms of an AP.

Sum of an AP Example 1 • Evaluate • Rather than calculating all of the terms, we will see that this is an AP and so we can use the formula for the sum of an AP.

Sum of an AP Example 1 • Solution: • Consider the first few terms: • This is an AP where n = 8, a = 4/3 and d = – 2/3 • Therefore,

Sum of an AP Example 2 • In an AP the sum of the first 10 terms is 50 and the 5 th term is three times the 2 nd term. Find the 1 st term and the sum of the first 20 terms. • We will find the 1 st term by solving simultaneous equations. This will also give us the common difference and we can then find the sum of the first 20 terms using the formula for Sn.

Sum of an AP Example 2 • Solution:

Sum of an AP Example 3 • Show that the terms of the following series are in arithmetic progression: • Then find the sum of the first 10 terms.

Sum of an AP Example 3 • Solution: • To show the terms are in arithmetic progression, we must show that there is a common difference.

Sum of an AP Example 3 • There is a common difference of ln 2 and so the series is in arithmetic progression. • a = ln 2 and d = ln 2. So the sum of the first 10 terms is:

Sum of an AP Example 4 • The sum of the first n terms of a sequence is given by n(n + 3). Find the fourth term of the sequence and show that the terms are in arithmetic progression. • Since we have the formula for the sum, then the sum of the first four terms subtract the sum of the first three terms will give us the fourth term itself. To show that the series is in arithmetic progression, we must show that the difference between a general term and the next term is a constant.

Sum of an AP Example 4 • Solution: Finding the fourth term:

Sum of an AP Example 4 • Solution: Show series is an AP: • Since an and an– 1 are general terms then this holds for all n. Since the difference between two general terms is a constant then this means the series is an AP.

Sum of an AP Example 5 • Find the least number of terms of the AP 1 + 3 + 5 + … that are required to make a sum exceeding 2000. • We need to establish the first term and the common difference to solve the resulting inequality in n, Sn > 2000.

Sum of an AP Example 5 • Solution:

Sum of an AP Example 6 • Find three numbers in AP whose sum is 15 and whose product is 80. • We need to consider three general terms in AP and then form simultaneous equations using their sum and their product.

Sum of an AP Example 6 • Solution: • Let the three general terms in AP be a, a + d and a + 2 d:

Sum of an AP Example 6 • Solving [1] and [3] gives two solutions: • a = 8 and d = – 3 • a = 2 and d = 3 • Substituting either of these pairs into a, a + d and a + 2 d gives the same 3 numbers. • Therefore three numbers in AP are 2, 5 and 8.

In summary • An arithmetic progression (AP) is a sequence where we add or subtract the same amount from term to term. • The nth term of an AP is: Tn = a + (n – 1)d • The sum of the first n terms of an AP is: • where a is the first term, d is the common difference and l is the last term.