Page 1 ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the common difference. Page 2 ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the common difference. 3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d = 4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: 4 1 ? ? ? n n a a The first term in the sequence would be a 1 which is sometimes just written as a. a = 3 Page 3 ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the common difference. 3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d = 4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: 4 1 ? ? ? n n a a The first term in the sequence would be a 1 which is sometimes just written as a. a = 3 3, 7, 11, 15, 19 … +4 +4 +4 +4 Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times. d = 4 ? ?d n a a n 1 ? ? ? Try this to get the 5th term. a = 3 ? ? 19 16 3 4 1 5 3 5 ? ? ? ? ? ? a Page 4 ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the common difference. 3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d = 4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: 4 1 ? ? ? n n a a The first term in the sequence would be a 1 which is sometimes just written as a. a = 3 3, 7, 11, 15, 19 … +4 +4 +4 +4 Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times. d = 4 ? ?d n a a n 1 ? ? ? Try this to get the 5th term. a = 3 ? ? 19 16 3 4 1 5 3 5 ? ? ? ? ? ? a Let’s look at a formula for an arithmetic sequence and see what it tells us. ? ? 1 4 ? n Subbing in the set of positive integers we get: 3, 7, 11, 15, 19, … What is the common difference? d = 4 you can see what the common difference will be in the formula We can think of this as a “compensating term”. Without it the sequence would start at 4 but this gets it started where we want it. 4n would generate the multiples of 4. With the - 1 on the end, everything is back one. What would you do if you wanted the sequence 2, 6, 10, 14, 18, . . .? ? ? 2 4 ? n Page 5 ARITHMETIC SEQUENCES These are sequences where the difference between successive terms of a sequence is always the same number. This number is called the common difference. 3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d = 4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: 4 1 ? ? ? n n a a The first term in the sequence would be a 1 which is sometimes just written as a. a = 3 3, 7, 11, 15, 19 … +4 +4 +4 +4 Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times. d = 4 ? ?d n a a n 1 ? ? ? Try this to get the 5th term. a = 3 ? ? 19 16 3 4 1 5 3 5 ? ? ? ? ? ? a Let’s look at a formula for an arithmetic sequence and see what it tells us. ? ? 1 4 ? n Subbing in the set of positive integers we get: 3, 7, 11, 15, 19, … What is the common difference? d = 4 you can see what the common difference will be in the formula We can think of this as a “compensating term”. Without it the sequence would start at 4 but this gets it started where we want it. 4n would generate the multiples of 4. With the - 1 on the end, everything is back one. What would you do if you wanted the sequence 2, 6, 10, 14, 18, . . .? ? ? 2 4 ? n Find the nth term of the arithmetic sequence when a = 6 and d = -2 If we use -2n we will generate a sequence whose common difference is -2, but this sequence starts at -2 (put 1 in for n to get first term to see this). We want ours to start at 6. We then need the “compensating term”. If we are at -2 but want 6, we’d need to add 8. ? ? 8 2 ? ? n Check it out by putting in the first few positive integers and verifying that it generates our sequence. 6, 4, 2, 0, -2, . . . Sure enough---it starts at 6 and has a common difference of -2Read More

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