Arithmetic ProgressionsIntroduction
Mehul is a carpenter in a village near Nashik, and a grape farmer has ordered a ladder for his vineyard. To keep it light enough to carry between the vines, the ladder gets narrower towards the top. The bottom rung is 45 cm long, and every rung is exactly 2 cm shorter than the rung just below it. There are eight rungs in all.
Before Mehul picks up his saw, he wants a cutting list: the length of every single rung.
Fig. 5.1 is live. You are Mehul's helper. The bottom rung is already cut — choose the length of each next rung until all eight are on the ladder.
Look at your cutting list: 45, 43, 41, 39, 37, 35, 33, 31. You never needed a measuring tape — every number came from the one before it by taking away 2. Once you know the rule behind a list of numbers, you can carry the list on as far as you like.
Patterns All Around You
Rules like Mehul's hide everywhere. Nature is full of them: the petals of a sunflower, the cells of a honeycomb, the rows of grains on a maize cob, the spirals on a pineapple and on a pine cone.
Everyday life has them too. Fill in the missing number in each list.
(i) Reena has just been selected for a job. Her starting salary is ₹8000 a month, and she gets a raise of ₹500 every year. Her monthly salaries (in ₹) in the 1st, 2nd, 3rd, … years are
8000,
Rule: add ₹500 every year — the same number each time.
(ii) Mehul's ladder (Fig. 5.1) gave the rung lengths (in cm)
45, 43 −2, 41 −2, 39 −2, 37 −2, 35 −2, 33 −2, 31 −2
(iii) A savings scheme makes money grow to times itself every 3 years. If ₹8000 is invested, the amounts (in ₹) after 3, 6, 9 and 12 years are
10000,
Rule: multiply by every 3 years — a multiplication, not an addition.
(iv) Squares of side 1, 2, 3, … units are cut into unit squares (Fig. 5.2). The numbers of unit squares are , that is,
1, 4 +3, 9 +5,
Rule: square the side. The gaps 3, 5, 7, … keep growing.
Fig. 5.2 — squares of side 1, 2, 3 and 4 units, cut into unit squares.
(v) Shakila drops ₹100 into her daughter's money box on her first birthday, and every year after that she puts in ₹50 more than the year before. The amounts (in ₹) on the 1st, 2nd, 3rd, 4th, … birthdays are
100, 150 +50, 200 +50,
Rule: add ₹50 every year.
(vi) A newborn pair of rabbits is too young to breed in its first month. From its second month on, it produces one new pair every month, and every new pair follows the same rule (Fig. 5.3). If no rabbit dies, the numbers of pairs at the start of the 1st, 2nd, 3rd, …, 6th months are
1, 1 +0, 2 +1, 3 +1, 5 +2,
Rule: add the two terms before. The gaps 0, 1, 1, 2, 3 keep changing.
Fig. 5.3 — one dot for each pair of rabbits: small dots are newborn pairs, large blue dots are pairs old enough to produce a new pair every month.
Did you know?
In the rabbit list 1, 1, 2, 3, 5, 8, … each number is the sum of the two before it. It is called the Fibonacci sequence, after the Italian mathematician
Three Kinds of Rules
Look back at the six lists. In some of them, every term comes from the one before it by adding a fixed number. In another, by multiplying by a fixed number. In another, the terms are the
Here are the six lists again, without their stories. Drag each one into the rule that builds it.
This chapter is about the first kind: lists in which you always add the same fixed number. You will learn to find any term of such a list without writing out all the terms before it, to add up its first terms in one line, and to use both skills on everyday problems.