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The fourth term of a geometric sequence is 48 - HSC - SSCE Mathematics Advanced - Question 21 - 2023 - Paper 1

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The fourth term of a geometric sequence is 48. The eighth term of the same sequence is $ rac{3}{16}$. Find the possible value(s) of the common ratio and the corres... show full transcript

Worked Solution & Example Answer:The fourth term of a geometric sequence is 48 - HSC - SSCE Mathematics Advanced - Question 21 - 2023 - Paper 1

Step 1

Let $a$ = first term and $r$ = common ratio

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Answer

We know the fourth and eighth terms of the geometric sequence. Therefore, we can express them in terms of aa and rr:

  1. The fourth term is given by: ar3=48ag1a r^3 = 48 ag{1}

  2. The eighth term is given by: ar7=316ag2a r^7 = \frac{3}{16} ag{2}

Step 2

Divide equation (2) by equation (1)

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Answer

Dividing equation (2) by equation (1), we get:

ar7ar3=31648\frac{a r^7}{a r^3} = \frac{\frac{3}{16}}{48}

This simplifies to:

r4=316×148=3768=1256r^4 = \frac{3}{16} \times \frac{1}{48} = \frac{3}{768} = \frac{1}{256}

Thus, we find:

r=12564=14  or  r=−14r = \sqrt[4]{\frac{1}{256}} = \frac{1}{4} \; \text{or} \; r = -\frac{1}{4}

Step 3

Find first term(s) corresponding to $r$

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Answer

For r=14r = \frac{1}{4}:

Using equation (1): a(14)3=48a \left( \frac{1}{4} \right)^3 = 48

This gives: a⋅164=48  ⟹  a=48×64=3072a \cdot \frac{1}{64} = 48 \implies a = 48 \times 64 = 3072

For r=−14r = -\frac{1}{4}:

Using the same equation: a(−14)3=48a \left( -\frac{1}{4} \right)^3 = 48

This gives: a⋅−164=48  ⟹  a=48⋅−64=−3072a \cdot -\frac{1}{64} = 48 \implies a = 48 \cdot -64 = -3072

Therefore, the possible first terms are 30723072 and −3072-3072.

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