Introduction
You must have observed that in nature, many things follow a certain pattern, such as the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the spirals on a pineapple and on a pine cone, etc.
We will now look for some patterns which occur in our day-to-day life. Some such examples are :
(i) Reena applied for a job and got selected. She has been offered a job with a starting monthly salary of Rs. 8000, with an annual increment of ₹ 500 in her salary. Her salary (in Rs.) for the 1st, 2nd, 3rd, . . . years will be, respectively 8000, 8500, 9000, . . . .
(ii) The lengths of the rungs of a ladder decrease uniformly by 2 cm from bottom to top. The bottom rung is 45 cm in length. The lengths (in cm) of the 1st, 2nd, 3rd, . . ., 8th rung from the bottom to the top are, respectively: 45, 43, 41, 39, 37, 35, 33, 31
(iii) In a savings scheme, the amount becomes
10000, 12500, 15625, 19531.25
(iv) The number of unit squares in squares with side 1, 2, 3, . . . units are, respectively
(v) Shakila puts Rs. 100 into her daughter’s money box when she was one year old and increased the amount by Rs. 50 every year. The amounts of money (in Rs.) in the box on the 1st, 2nd, 3rd, 4th, . . . birthday were 100, 150, 200, 250, . . ., respectively.
(vi) A pair of rabbits are too young to produce in their first month. In the second, and every subsequent month, they produce a new pair. Each new pair of rabbits produce a new pair in their second month and in every subsequent month (see Fig. 5.3). Assuming no rabbit dies, the number of pairs of rabbits at the start of the 1st, 2nd, 3rd, . . ., 6th month, respectively are : 1, 1, 2, 3, 5, 8
In the first month, the rabbits are very small and can’t do much – but they grow very quickly.
After one month, the rabbits are grown up and can start mating…
… and after another month, they will give birth to their first pair of kids. You now have two pairs of rabbits.
In the next month, your pair of rabbits will give birth to another couple. Meanwhile, the first pair of kids have grown up. You now have three pairs in total.
In the fifth month, your original pair of rabbits will give birth to a new pair. At the same time, their first pair of kids is now old enough to give birth to grandchildren. You now have five pairs of rabbits.
In the sixth month, there are three more couples that give birth: the original one, as well as their first two pairs or kids.
In all the examples above, we observe some patterns. In some, we find that the succeeding terms are obtained by adding a fixed number, in other by multiplying with a fixed number, in another we find that they are squares of consecutive numbers, and so on.
In this chapter, we shall discuss one of these patterns in which succeeding terms are obtained by adding a fixed number to the preceding terms. We shall also see how to find their nth terms and the sum of n consecutive terms, and use this knowledge in solving some daily life problems.