Question

6 the table shows a linear pattern. a) predict the value of term 31. each term is? less than the previous term. 31 - 1 =? ? ×? =? 48.4 -? =? the value of term 31 is?. determine the number of times? is subtracted from term calculate? times?. subtract? from 48.4, the value of term 1. b) predict the term number that corresponds to the value 38.4. 48.4 - 38.4 =? ? ÷? =? 1 +? =? the term number that corresponds to the value 38.4 is?. subtract 38.4 from the value of term 1. determine the number of times? was subtracted to make add this number to term 1. c) what is the equation for the pattern? the value of term 1 is (? - 1 ×?). the value of term 2 is (? - 2 ×?). the equation for the pattern is n =? -? t. term number (t) term value (n) 1 48.4 2 48.2 3 48.0 4 47.8

Explanation:

Step1: Find the common difference

$48.4 - 48.2 = 0.2$

Step2: Calculate term difference

$31 - 1 = 30$

Step3: Total decrease over 30 terms

$30 \times 0.2 = 6$

Step4: Find term 31 value

$48.4 - 6 = 42.4$
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Step5: Total decrease for value 38.4

$48.4 - 38.4 = 10$

Step6: Number of subtractions

$\frac{10}{0.2} = 50$

Step7: Find the term number

$1 + 50 = 51$
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Step8: Identify base and coefficient

Base value (when $t=0$): $48.4 + 0.2 = 48.6$; coefficient: $0.2$

Step9: Form the linear equation

$N = 48.6 - 0.2t$

Answer:

Part a)

Each term is $\boldsymbol{0.2}$ less than the previous term.
$31 - 1 = \boldsymbol{30}$
$\boldsymbol{30} \times \boldsymbol{0.2} = \boldsymbol{6}$
$48.4 - \boldsymbol{6} = \boldsymbol{42.4}$
The value of term 31 is $\boldsymbol{42.4}$.

Part b)

$48.4 - 38.4 = \boldsymbol{10}$
$\boldsymbol{10} \div \boldsymbol{0.2} = \boldsymbol{50}$
$1 + \boldsymbol{50} = \boldsymbol{51}$
The term number that corresponds to the value 38.4 is $\boldsymbol{51}$.

Part c)

The value of term 1 is $(\boldsymbol{48.6} - 1 \times \boldsymbol{0.2})$.
The value of term 2 is $(\boldsymbol{48.6} - 2 \times \boldsymbol{0.2})$.
The equation for the pattern is $N = \boldsymbol{48.6} - \boldsymbol{0.2}t$.