Exercise practice 1: Algebra Play
Algebra Play22 min readLesson 3 of 6
3.1 Figure it Out: pyramid tops
3.1.1 FIO 1, Q1: top of a three-row pyramid from the bottom row
Statement: without building the whole pyramid, find the top for the bottom rows (4, 13, 8), (7, 11, 3) and (10, 14, 25).
Key step: for three rows the top is a + 2b + c.
- 4 + 2 × 13 + 8 = 4 + 26 + 8 = 38.
- 7 + 2 × 11 + 3 = 7 + 22 + 3 = 32.
- 10 + 2 × 14 + 25 = 10 + 28 + 25 = 63.
Answer: 38, 32 and 63.
3.1.2 FIO 1, Q2: expression for a four-row pyramid
Key step: label the bottom row a, b, c, d. The next row is a + b, b + c, c + d; the third row is a + 2b + c and b + 2c + d; the top is their sum.
Answer: top = a + 3b + 3c + d.
3.1.3 FIO 1, Q3: four-row pyramid tops
Statement: find the top for bottom rows (8, 19, 21, 13), (7, 18, 19, 6) and (9, 7, 5, 11).
Key step: use a + 3b + 3c + d.
- 8 + 3 × 19 + 3 × 21 + 13 = 8 + 57 + 63 + 13 = 141.
- 7 + 3 × 18 + 3 × 19 + 6 = 7 + 54 + 57 + 6 = 124.
- 9 + 3 × 7 + 3 × 5 + 11 = 9 + 21 + 15 + 11 = 56.
Answer: 141, 124 and 56.
3.1.4 FIO 1, Q4: first three Fibonacci numbers in a three-row pyramid
Statement: put 1, 2, 3 in the bottom row of a three-row pyramid, fill it, and describe the numbers.
Key step: the middle row is 1 + 2 = 3 and 2 + 3 = 5. The top is 3 + 5 = 8.
Answer: the pyramid contains 1, 2, 3, 5 and 8, all Virahāṅka-Fibonacci numbers; the top is 8, which is the fifth Fibonacci number.
3.1.5 FIO 1, Q5: longer Fibonacci pyramids
(i) A four-row pyramid with bottom row 1, 2, 3, 5: middle rows are 3, 5, 8 and 8, 13; the top is 21. (ii) A 29-row pyramid with the first 29 Fibonacci numbers: every entry is a Fibonacci number, and the top is the 57th Fibonacci number.
Answer: the top of an n-row pyramid fed with the first n Fibonacci numbers is the (2n − 1)th Fibonacci number; for n = 3 this is 8 and for n = 4 it is 21.
3.1.6 FIO 1, Q6: general statement
Statement: suppose an n-row pyramid is fed with the first n Virahāṅka-Fibonacci numbers along its base. What can be said about all the entries and about the final top value?
Answer: every number that appears is a Virahāṅka-Fibonacci number. The top is the (2n − 1)th Fibonacci number, because each upward step adds neighbouring Fibonacci numbers and the indices grow by one per row while the ends keep the outer index.
3.2 Checking a pyramid formula
To trust the formula, build one pyramid by hand for the bottom row 8, 19, 21, 13. The second row is 8 + 19 = 27, then 19 + 21 = 40, then 21 + 13 = 34. The third row is 27 + 40 = 67 and 40 + 34 = 74. The top is 67 + 74 = 141, which matches the formula 8 + 3 × 19 + 3 × 21 + 13 = 141. The coefficients count how often each bottom number contributes to the top. Building one example by hand and then using the formula is the safest habit for exams, because the hand check catches a wrong coefficient or a copied number. For the three-row case the coefficients are 1, 2 and 1, so the two end numbers are counted once and the middle number twice; knowing that pattern makes the four-row coefficients easy to remember.
3.2.1 Choosing the fastest method
Before calculating, decide whether the short formula or a hand-built pyramid is safer. The formula is faster once the coefficients are known, while building the pyramid is a reliable check for a single small case. Doing both on one example confirms the formula before you trust it on the rest, and it also shows why the middle entries are counted more than once.
3.3 Figure it Out: the largest product
3.3.1 FIO 2, Q1: digits 1, 3, 7
Statement: place 1, 3 and 7 into a two-digit number times a one-digit number, using each digit once, to make the largest product.
Key step: the largest digit is the multiplier and the other two digits are written in decreasing order: 73 × 1. Compare with the other candidates: 31 × 7 = 217, 71 × 3 = 213, 37 × 1 = 37, 17 × 3 = 51, 13 × 7 = 91, 73 × 1 = 73.
Answer: 31 × 7 = 217 is the largest.
3.3.2 FIO 2, Q2: digits 3, 5, 9
Statement: same task with digits 3, 5 and 9.
Key step: use the general rule: 53 × 9. The other candidates are 35 × 9 = 315, 39 × 5 = 195, 59 × 3 = 177, 93 × 5 = 465, 95 × 3 = 285 and 53 × 9 = 477.
Answer: 53 × 9 = 477 is the largest.