Predicting What Comes Next: Sequences and Progressions | EOT

Question 5

Find a GP for which the sum of the first two terms is −4-4 and the fifth term is 4 times the third term.

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Solution
Understand the Question
  • Let the first term of the Geometric Progression (GP) be aa and the common ratio be rr.
  • The nthn^{\text{th}} term of a GP is given by the formula an=arn−1a_n = a r^{n-1}.
  • We are given two conditions:
    1. The sum of the first two terms is −4-4: a1+a2=−4a_1 + a_2 = -4
    2. The fifth term is 44 times the third term: a5=4a3a_5 = 4 a_3
  • We use these conditions to solve for rr and aa, which will give us the required GP(s).

Step 1 · Set up equations from given conditions

Let the first term be aa and the common ratio be rr.

From the first condition, the sum of the first two terms is −4-4:

a+ar=−4  ⟹  a(1+r)=−4…(1)a + ar = -4 \implies a(1 + r) = -4 \quad \dots (1)

From the second condition, the fifth term is 44 times the third term:

a5=4a3a_5 = 4 a_3

Using the formula an=arn−1a_n = a r^{n-1}:

ar4=4(ar2)…(2)ar^4 = 4(ar^2) \quad \dots (2)

Step 2 · Find the common ratio rr

Since a≠0a \neq 0 and r≠0r \neq 0 (otherwise the sum of terms would be 00, not −4-4), divide both sides of equation (2)(2) by ar2ar^2:

ar4ar2=4r2=4r=±2\begin{aligned} \frac{ar^4}{ar^2} &= 4 \\[0.6em] r^2 &= 4 \\[0.6em] r &= \pm 2 \end{aligned}

Step 3 · Determine the first term and the GP for each value of rr

Case 1: When r=2r = 2

Substitute r=2r = 2 into equation (1)(1):

a(1+2)=−43a=−4a=−43\begin{aligned} a(1 + 2) &= -4 \\[0.6em] 3a &= -4 \\[0.6em] a &= -\frac{4}{3} \end{aligned}

The GP is a,ar,ar2,ar3,…a, ar, ar^2, ar^3, \dots, which gives:

−43,−83,−163,…-\frac{4}{3}, -\frac{8}{3}, -\frac{16}{3}, \dots

Case 2: When r=−2r = -2

Substitute r=−2r = -2 into equation (1)(1):

a(1−2)=−4−a=−4a=4\begin{aligned} a(1 - 2) &= -4 \\[0.6em] -a &= -4 \\[0.6em] a &= 4 \end{aligned}

The GP is a,ar,ar2,ar3,…a, ar, ar^2, ar^3, \dots, which gives:

4,−8,16,−32,…4, -8, 16, -32, \dots
Answer

−43,−83,−163,…-\dfrac{4}{3}, -\dfrac{8}{3}, -\dfrac{16}{3}, \dots or 4,−8,16,−32,…4, -8, 16, -32, \dots

Common Mistakes
  • Missing the Negative Root: Writing r=2r = 2 and ignoring r=−2r = -2 when solving r2=4r^2 = 4. This leads to missing the second valid GP.
  • Incorrect Formula for Terms: Confusing an=arn−1a_n = ar^{n-1} with an=arna_n = ar^n.

More questions in EOT

Q1

Find the 31st31^{\text{st}} term of an AP whose 11th11^{\text{th}} term is 38 and 16th16^{\text{th}} term is 73.

Q2

Determine the AP whose third term is 16 and whose 7th7^{\text{th}} term exceeds the 5th5^{\text{th}} term by 12.

Q3

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Q4

How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)

Q5

Find a GP for which the sum of the first two terms is −4-4 and the fifth term is 4 times the third term.

Q6

Find all possible ways of expressing 100100 as the sum of consecutive natural numbers.

Q7

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd2^{\text{nd}} hour, 4th4^{\text{th}} hour and nthn^{\text{th}} hour?

Q8

The sum of the 4th4^{\text{th}} and 8th8^{\text{th}} terms of an AP is 24 and the sum of the 6th6^{\text{th}} and 10th10^{\text{th}} terms is 44. Find the first three terms of the AP.

Q9

Find the smallest value of nn such that the sum of the first nn natural numbers is greater than 1,000.

Q10

Which term of the GP: 2,8,32,…2, 8, 32, \dots is 131072131072? Write the explicit formula as well as the recursive formula for the nthn^{\text{th}} term.

Q11

The sum of the first three terms of a GP is 1312\dfrac{13}{12} and their product is −1-1. Find the common ratio and the terms.

Q12

If the 4th4^{\text{th}}, 10th10^{\text{th}} and 16th16^{\text{th}} terms of a GP are xx, yy and zz respectively, prove that x,y,zx, y, z are in GP.

Q13

The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

Q14

Suppose P1=1P_1 = 1, P2=2P_2 = 2 and for n>2n > 2, Pn=P1+P2+⋯+Pn−1+1P_n = P_1 + P_2 + \dots + P_{n-1} + 1. Find the values of P1,P2,…,P8P_1, P_2, \dots, P_8. Can you find a simpler recursive formula for PnP_n? Can you give an explicit formula?

Q15

Suppose W1=1W_1 = 1, W2=2W_2 = 2 and for n>2n > 2, Wn=W1+W2+⋯+Wn−2+2W_n = W_1 + W_2 + \dots + W_{n-2} + 2. Find the values of W1,W2,…,W8W_1, W_2, \dots, W_8. Do you recognise this sequence?

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