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For each arithmetic progression (AP), find the first term and common difference: i) 3, 1, -1, -3, ... ii) -5, -1, 3, 7, ... iii) 1/3, 5/3, 9/3, 13/3, ... iv) 0.6, 1.7, 2.8, 3.9, ...

Mathematics

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Answers

4 parts
i

a=3a = 3

d=−2d = -2

ii

a=−5a = -5

d=4d = 4

iii

a=13a = \frac{1}{3}

d=43d = \frac{4}{3}

iv

a=0.6a = 0.6

d=1.1d = 1.1

Step-by-step solution

8 steps

Here are the first term and common difference for each AP:

Part i· 2 steps

3,1,−1,−3,…3, 1, -1, -3, \dots

  1. Identify the first term.

    The first term is the first number in the sequence.

    a=3a = 3
  2. Calculate the common difference.

    The common difference dd is found by subtracting any term from its succeeding term.

    d=a2−a1=1−3=−2d = a_2 - a_1 = 1 - 3 = -2 d=a3−a2=−1−1=−2d = a_3 - a_2 = -1 - 1 = -2

    The first term is a=3\boxed{a = 3} and the common difference is d=−2\boxed{d = -2}.

Part ii

a=−5a = -5

−5,−1,3,7,…-5, -1, 3, 7, \dots

  1. Identify the first term.

    The first term is the first number in the sequence.

    a=−5a = -5
  2. Calculate the common difference.

    d=a2−a1=−1−(−5)=−1+5=4d = a_2 - a_1 = -1 - (-5) = -1 + 5 = 4 d=a3−a2=3−(−1)=3+1=4d = a_3 - a_2 = 3 - (-1) = 3 + 1 = 4

    The first term is a=−5\boxed{a = -5} and the common difference is d=4\boxed{d = 4}.

Part iii

a=13a = \frac{1}{3}

13,53,93,133,…\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \dots

  1. Identify the first term.

    The first term is the first number in the sequence.

    a=13a = \frac{1}{3}
  2. Calculate the common difference.

    d=a2−a1=53−13=43d = a_2 - a_1 = \frac{5}{3} - \frac{1}{3} = \frac{4}{3} d=a3−a2=93−53=43d = a_3 - a_2 = \frac{9}{3} - \frac{5}{3} = \frac{4}{3}

    The first term is a=13\boxed{a = \frac{1}{3}} and the common difference is d=43\boxed{d = \frac{4}{3}}.

Part iv

a=0.6a = 0.6

0.6,1.7,2.8,3.9,…0.6, 1.7, 2.8, 3.9, \dots

  1. Identify the first term.

    The first term is the first number in the sequence.

    a=0.6a = 0.6
  2. Calculate the common difference.

    d=a2−a1=1.7−0.6=1.1d = a_2 - a_1 = 1.7 - 0.6 = 1.1 d=a3−a2=2.8−1.7=1.1d = a_3 - a_2 = 2.8 - 1.7 = 1.1

    The first term is a=0.6\boxed{a = 0.6} and the common difference is d=1.1\boxed{d = 1.1}.

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