Sequences - Term-to-Term Rules and Patterns

Find how much is added to get from one term to the next.

The terms go down by the same amount each time, and the sequence can carry on below zero.

Work out the difference between terms that are next to each other.

Use two terms that are next to each other to find the number added each time.

Write out the terms one at a time, starting with the 1st term.

Write out the terms one at a time, starting with the 1st term.

1. 1st term: 400
2. 2nd term: 400\div2=200
3. 3rd term: 200\div2=100
4. 4th term: 100\div2=50
5. 5th term: 50\div2=25

The 4th term comes just before the 5th term, so do the opposite of the rule.

1. To get from the 4th term to the 5th term, you subtract 6.
2. So to go back from the 5th term to the 4th term, add 6: 17+6=23.
3. Check: 23-6=17.

Count the tiles in each pattern and see how many are added each time.

1. Pattern 1 has 5 tiles, pattern 2 has 9 tiles and pattern 3 has 13 tiles.
2. Each new pattern adds one tile to the end of each of the 4 arms, so 4 more tiles each time.
3. Pattern 4 needs 13+4=17 tiles.

Count the sticks in each pattern, then find the difference between patterns that are next to each other.

1. Pattern 1 has 4 sticks, pattern 2 has 7 sticks and pattern 3 has 10 sticks.
2. 7-4=3 and 10-7=3, so the term-to-term rule is add 3.
3. Each new square shares one side with the square before it, so it needs only 3 more sticks.

Use the rule on the last term you know: multiply by 2 first, then subtract 3.

Part (a)

1. Check the rule: 4\times2-3=5 and 5\times2-3=7.
2. 4th term: 7\times2-3=14-3=11.

Part (b)

1. 5th term: 11\times2-3=22-3=19.

Count how many times you add the same number to get from the 4th term to the 10th term.

1. The term-to-term rule is add 7.
2. From the 4th term to the 10th term is 10-4=6 steps.
3. The difference is 6\times7=42.
4. Check: the 10th term is 29+42=71, and 71-29=42.

Count the grey tiles in each pattern and see how many more are needed each time.

Part (a)

1. Pattern 1 has 8 grey tiles, pattern 2 has 10 and pattern 3 has 12.
2. Each new pattern needs 2 more grey tiles: one more on the top row and one more on the bottom row.
3. Pattern 4 has 14 and pattern 5 has 16 grey tiles.

Part (b)

1. Each pattern has 2 more grey tiles than the one before.
2. From 8 grey tiles to 30 grey tiles is 30-8=22 more, which is 22\div2=11 steps.
3. So it is pattern number 1+11=12.
4. Check: pattern 12 has 12 white tiles in a row, with 14 grey tiles above, 14 below and 1 at each end: 14+14+2=30.

Week 8 is not 8 increases after week 1. Count the increases carefully.

1. From week 1 to week 8 there are 8-1=7 increases of 1.5 km.
2. 7\times1.5=10.5 km.
3. In week 8 he runs 3+10.5=13.5 km.

Work backwards from the 6th term, doing the opposite of the rule each time.

1. To go backwards, subtract 7 each time.
2. 5th term: 43, 4th term: 36, 3rd term: 29, 2nd term: 22, 1st term: 15.
3. From the 6th term back to the 1st term is 5 steps, and 50-5\times7=15.

Write out some terms of the sequence. What do they all have in common?

1. The sequence is 5, 9, 13, 17, \ldots, 53, 57, 61, \ldots
2. It starts with an odd number and adding 4 (an even number) always gives another odd number.
3. So every number in the sequence is odd, and 58 is even: 58 is not in the sequence.
4. The first reason is wrong: 5, 9 and 13 are not in the 4 times table either.

Count how many steps there are from the 2nd term to the 5th term.

1. From the 2nd term to the 5th term is 3 steps, and the terms go up by 26-11=15.
2. So the number added each time is 15\div3=5.
3. From the 5th term to the 8th term is another 3 steps: 26+3\times5=41.
4. The sequence is 6, 11, 16, 21, 26, 31, 36, 41.

Undo the rule: the last thing done was adding 2, so undo that first.

1. To go backwards, undo the rule in reverse order: subtract 2, then divide by 3.
2. 2nd term: 44-2=42, then 42\div3=14.
3. 1st term: 14-2=12, then 12\div3=4.
4. Check: 4\times3+2=14 and 14\times3+2=44.

Find a rule linking the pattern number to the number of tiles.

1. Pattern 1 uses 4 tiles. Each next pattern adds one tile at each end of the top row and one at the bottom: 3 extra tiles each time.
2. After making pattern 1, Kofi has 80-4=76 tiles left for these increases.
3. 76\div3=25 remainder 1. He can make 25 complete increases; the spare tile is not enough for another increase.
4. Starting at pattern 1 and increasing 25 times gives pattern 1+25=26.
5. Check: pattern 26 needs 4+25\times3=79 tiles. Pattern 27 would need 79+3=82, more than 80. The largest possible pattern number is 26.

Both rules give 3, 6, 12 at the start, but they carry on differently. Use each rule one step at a time.

Part (a)

1. 3, 6, 12, 24, 48
2. Alfie's 5th term is 48.

Part (b)

1. Bea adds 3, then 6, then 9, then 12.
2. 4th term: 12+9=21. 5th term: 21+12=33.
3. Bea's 5th term is 33.

Work out how much the gap between their savings changes each week.

1. At the start, Ellie has 50-14=36 pounds more than Sam.
2. Each week Sam saves 12-8=4 pounds more than Ellie, so the gap closes by 4 pounds a week.
3. After 36\div4=9 weeks they have the same: Ellie 50+9\times8=122, Sam 14+9\times12=122.
4. At the end of week 10: Ellie has 130 pounds and Sam has 134 pounds.
5. Sam first has more money at the end of week 10.