Note 5 Arithmetic Sequences n n An arithmetic

Note 5: Arithmetic Sequences n n An arithmetic sequence is where a number is added to get the next term The number is called the common difference The formula for an arithmetic sequence is: tn = a + (n – 1)d a = first term d = common difference tn = the nth term

Example 1 Calculate the 24 th term in the arithmetic sequence 2, 6, 10, 14, … a= 2 d= 4 tn = a + (n – 1)d t 24 = 2 + (24 – 1)4 = 94 Page 105 Exercise 13. 1 Q 1 -5

Sometimes you may not be asked for a term, but other parts of the formula, requiring rearranging and solving: Example 2 In the sequence 2, 4, 6, 8, …. what is term 644? a= 2 d= 2 tn = a + (n – 1)d 644 = 2 + (n – 1)x 2 644 = 2 + 2 n - 2 644 = 2 n n = 322

Example 3 Find the formula for the nth term of the sequence 5, 9, 13, 17, … a= 5 d= 4 tn = a + (n – 1)d tn = 17 + (n – 1)4 = 17 + 4 n - 4 = 4 n + 13

Example 4 An arithmetic sequence has a common difference of 5. Find the first term of the sequence if t 17 = 107 a= ? d= 5 n = 17 tn = a + (n – 1)d t 17 = a + (17 – 1)x 5 107 = a + 90 a = 17

Example 5 The 3 rd term of a sequence is 12 and the 7 th term is 20. Find the difference and the first term tn = a + (n – 1)d t 3 = 12 = a + (3 – 1)d 12 = a + 2 d t 7 = 20 = a + (3 – 1)d 20 = a + 6 d Solving simultaneous equations on calculator: EQUA – F 1( 2 unknowns ) a = 8 and d = 2

Example 6: Victoria is training for a cycle race. She increases her daily distance steadily through the month. On the 8 th day she cycled 46 km and on the 21 st day she cycled 72 km. Find her starting distance and the daily increase. a=? d=? t 8 = a + (8 - 1)d and t 21 = a + (21 - 1)d 46 = a + 7 d 72 = a + 20 d Solving simultaneous equations on calculator: EQUA – F 1( 2 unknowns ) a = 32 and d = 2, Starting distance is 32 km and daily increase is 2 km

Page 105 Exercise 13. 1 Q 6 -9