What are the next three numbers and the rule for the following sequences: c) 78, 67, 62, ... e) 3, 9, 27…

Question

What are the next three numbers and the rule for the following sequences: c) 78, 67, 62, ... e) 3, 9, 27, ... f) 1, 3, 10, 15, ...

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the sequences and the verification of your activity answers:

Mental Maths

c) 78, 67, 62

Find the differences between consecutive terms.
$67 - 78 = -11$ $62 - 67 = -5$
Identify the pattern in the differences.
The pattern is subtracting 11, then subtracting 5, and this pattern repeats.

Apply the rule to find the next three numbers.

$62 - 11 = 51$  
$51 - 5 = 46$  
$46 - 11 = 35$  
Next three numbers: $\boxed{\text{51, 46, 35}}$  
Rule: $\boxed{\text{Subtract 11, then subtract 5, alternating}}$

e) 3, 9, 27

Find the ratio between consecutive terms.
$\frac{9}{3} = 3$ $\frac{27}{9} = 3$
Identify the rule.
Each term is multiplied by 3 to get the next term.

Apply the rule to find the next three numbers.

$27 \times 3 = 81$  
$81 \times 3 = 243$  
$243 \times 3 = 729$  
Next three numbers: $\boxed{\text{81, 243, 729}}$  
Rule: $\boxed{\text{Multiply by 3}}$

f) 1, 3, 10, 15

Find the differences between consecutive terms.
$3 - 1 = 2$ $10 - 3 = 7$ $15 - 10 = 5$
The differences (2, 7, 5) do not follow a simple arithmetic or geometric pattern.
Given the context of "Mental Maths" problems, it is highly probable that this sequence was intended to be 1, 3, 6, 10, 15 (triangular numbers). Assuming this common pattern:
The differences would be:
$3 - 1 = 2$
$6 - 3 = 3$
$10 - 6 = 4$
$15 - 10 = 5$
Apply this assumed rule (add consecutive integers starting from 2) to find the next three numbers.
$15 + 6 = 21$ $21 + 7 = 28$ $28 + 8 = 36$ Next three numbers: $\boxed{\text{21, 28, 36}}$ Rule: $\boxed{\text{Add consecutive integers starting from 2}}$
Activity
- a) Describe the pattern in words. Your answer: "the pattern grows by adding even odd number" The pattern shows the number of dots as $

What are the next three numbers and the rule for the following sequences: c) 78, 67, 62, ... e) 3, 9, 27, ... f) 1, 3, 10, 15, ...

Step-by-step solution

Find the differences between consecutive terms.

Identify the pattern in the differences.

Apply the rule to find the next three numbers.

Find the ratio between consecutive terms.

Identify the rule.

Apply the rule to find the next three numbers.

Find the differences between consecutive terms.

The differences (2, 7, 5) do not follow a simple arithmetic or geometric pattern.

Apply this assumed rule (add consecutive integers starting from 2) to find the next three numbers.

Still stuck on this one?

More Mathematics Questions