Question 10 - End-of-Chapter Exercises - Chapter 8 - Predicting What Comes Next: Exploring Sequences & Progress

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Question 10 Which term of the GP: 2, 8, 32… is 131072 ? Write the explicit formula as well as the recursive formula for the 𝑛^"th " term. Given G.P., 2, 8, 32, ...upto n terms We know that an = arn – 1 Here, First term = a = 2 Common ratio = r = 8/2 = 4 nth term = an = 131072 We need to find n Now an = arn – 1 Putting values 131072 = 2 × 4n – 1 131072/2 = 4n – 1 65536 = 4n – 1 4n – 1 = 65536 Writing 65536 as power of 4 4n – 1 = 48 Comparing powers n – 1 = 8 n = 8 + 1 n = 9 Hence, 131072 is the 9th term of the G.P Now, we are asked Write the explicit formula as well as the recursive formula for the nth term Explicit formula An explicit formula allows you to calculate any term in a sequence directly, just by knowing its position number (n) Thus, explicit formula for nth term of GP is an = arn – 1 Putting a = 2, r = 4 Our explicit formula is an = 2 × 4n – 1 Recursive formula A recursive formula tells you how to find the next term in a sequence by doing something to the previous term. It always requires two pieces of information: The starting point (usually the first term, a) The rule to get from one term (an–1) to the next term (an) For our GP, our recursive formula is an = r × an–1 Putting r = 4 an = 4 × an–1 Where a1 = a = 2 Note: In recursive formula, we need the first term as well as the rule. Only rule is not a recursive formula