Mathematics Year 7: Sequences of rational numbers and the general term
Mathematics, Year 7 (Portugal): With negatives, decimals and fractions in the list, the rule still comes from the jump between terms, and the general term gives any term straight from its position.
The rule, and the general term
A sequence of rational numbers can hold negatives, decimals and fractions. In −7; −4.5; −2; 0.5… each term is the previous one plus 2.5: that is the rule, said in words. The rule can also give a term from its position, the general term. From order 1 to order n there are n − 1 jumps of 2.5, so the term is −7 + 2.5 × (n − 1), which simplifies to 2.5 × n − 9.5. Test it: n = 1 gives −7 and n = 4 gives 0.5. A rule can be said in words or in symbols, and it only counts if it fits every term you know.
One sequence, four faces
The same sequence can be shown as a drawing, a table, words and a formula, and they must tell the same story. Take hexagons (side 1) joined in a row: figure n has n hexagons, and its perimeter is 6, 10, 14, 18… Words: each new hexagon adds 4. Drawing: n hexagons have 2n sides on top, 2n below and 1 at each end, so 4n + 2. Counting differently: 6n sides minus 2 for each of the n − 1 joins, so 6n − 2(n − 1). Both give 22 for n = 5 (6 × 5 − 2 × 4 = 22 and 4 × 5 + 2 = 22). Different formulas, one sequence.
Terms before, and far after, the ones shown
A table shows only the orders 4 to 7: 0.5; 3; 5.5; 8, and each term is 2.5 above the previous one. Order 1 is 3 jumps back: 0.5 − 3 × 2.5 = 0.5 − 7.5 = −7. So the general term is −7 + 2.5 × (n − 1) = 2.5 × n − 9.5. Order 20: 2.5 × 20 − 9.5 = 50 − 9.5 = 40.5. Check by jumps: −7 + 19 × 2.5 = −7 + 47.5 = 40.5, the same. Order 100: 250 − 9.5 = 240.5. Is 33 a term? From −7 to 33 is 40, and 40 ÷ 2.5 = 16 jumps exactly, so 33 has order 17. Is 20? 27 ÷ 2.5 = 10.8, not whole: no.
Zero is not the end
Sequence 3/2; 1; 1/2; 0… and the rule is to subtract 1/2. Many people stop at 0, or turn round and write 1/2; 1; 3/2, as if the numbers bounced off zero. That worked in earlier years, when sequences of numbers ended at 0. Now there are negative numbers, and the rule does not change when you cross zero: 0 − 1/2 = −1/2, then −1, then −3/2. The general term is (4 − n)/2: n = 1 gives 3/2 and n = 7 gives −3/2. The jump is the same on both sides of zero.
Falling temperatures and spreadsheets
A temperature that falls 1.5 °C every hour from 4 °C gives 4; 2.5; 1; −0.5; −2… a sequence of rational numbers that crosses zero. Its general term is 5.5 − 1.5 × n: n = 1 gives 4 and n = 5 gives 5.5 − 7.5 = −2. A spreadsheet does this work for you: type the first term in one cell, and in the cell below say "the one above plus 2.5" (or minus 1.5); drag down and it lists as many terms as you like. You supply the rule, the machine repeats it, and you still check the first terms by hand.
Test what you learned
The sequence 1.2; 0.7; 0.2; −0.3… follows a rule. Which one?
- a) Add 0.5 to the previous term
- b) Subtract 0.7 from the previous term
- c) Subtract 0.5 from the previous term
- d) Multiply the previous term by 0.5
Correct answer: c) Subtract 0.5 from the previous term — Right: 1.2 − 0.5 = 0.7, 0.7 − 0.5 = 0.2, 0.2 − 0.5 = −0.3. The same jump every time, even past zero.
A sequence starts 2; 0.5; −1; −2.5… (each term is 1.5 below the previous one). What is the term of order 12?
- a) −14.5
- b) −16
- c) −18
- d) 16.5
Correct answer: a) −14.5 — Right: from order 1 to order 12 there are 11 jumps, so 2 − 1.5 × 11 = 2 − 16.5 = −14.5.
A sequence starts 1.5; 1; 0.5; 0… and the rule is to subtract 0.5 each time. What is the term of order 7?
- a) 1.5
- b) −2
- c) It does not exist: the sequence ends at 0
- d) −1.5
Correct answer: d) −1.5 — Right: 1.5 − 0.5 × 6 = 1.5 − 3 = −1.5. The sequence goes 0, −0.5, −1, −1.5: the same jump on both sides of zero.
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