<span>The correct answer is 9.8 ft.
Explanation:
This is a geometric sequence, which follows the explicit formula
</span>
<span>
where g</span>₁<span> is the first term, r is the common ratio and n is the term number.
We know that the third term is 18; this gives us 18=g</span>₁×<span>r</span>³⁻¹<span> or 18=g</span>₁×<span>r</span>²<span>.
We also know the seventh term is 8, which gives us 8=g</span>₁×<span>r</span>⁷⁻¹<span> or 8=g</span>₁×<span>r</span>⁶<span>.
Solving for g</span>₁<span> in the third term gives us g</span>₁<span>=18/r</span>²<span>, and solving for g</span>₁<span> in the seventh term gives us g</span>₁<span>=8/r</span>⁶<span>. They both equal g</span>₁<span> so we set them equal to each other:
18/r</span>²<span> = 8/r</span>⁶<span>.
Multiply both sides by r</span>⁶<span>, which gives us
18r</span>⁶<span>/r</span>²<span> = 8.
Using our properties of exponents, we have 18r</span>⁴<span> = 8. Divide both sides by 18, which gives us
r</span>⁴<span>=8/18.
We can find the fourth root by taking the square root twice:
taking the square root gives us r</span>²<span> = </span>√<span>8/</span>√<span>18.
Simplifying </span>√<span>8 we get 2</span>√<span>2, and simplifying </span>√<span>18 gives us 3</span>√<span>2; we now have
r</span>²<span>=2</span>√<span>2)/3</span>√<span>2.
The </span>√<span>2 will cancel, leaving r</span>²<span>=2/3. Taking the square root again, we have
r=</span>√<span>2/</span>√<span>3; simplifying this gives us r=</span>√<span>6/3.
We can now work backward to find the sixth term using the seventh one; Divide 8 by </span>√<span>6/3. Dividing by a fraction means multiplying by the reciprocal, so we multiply 8 by 3/</span>√<span>6; this gives us 24/</span>√<span>6, and in a calculator that gives us 9.8 ft.</span>
Explanation:
This is a geometric sequence, which follows the explicit formula
</span>
where g</span>₁<span> is the first term, r is the common ratio and n is the term number.
We know that the third term is 18; this gives us 18=g</span>₁×<span>r</span>³⁻¹<span> or 18=g</span>₁×<span>r</span>²<span>.
We also know the seventh term is 8, which gives us 8=g</span>₁×<span>r</span>⁷⁻¹<span> or 8=g</span>₁×<span>r</span>⁶<span>.
Solving for g</span>₁<span> in the third term gives us g</span>₁<span>=18/r</span>²<span>, and solving for g</span>₁<span> in the seventh term gives us g</span>₁<span>=8/r</span>⁶<span>. They both equal g</span>₁<span> so we set them equal to each other:
18/r</span>²<span> = 8/r</span>⁶<span>.
Multiply both sides by r</span>⁶<span>, which gives us
18r</span>⁶<span>/r</span>²<span> = 8.
Using our properties of exponents, we have 18r</span>⁴<span> = 8. Divide both sides by 18, which gives us
r</span>⁴<span>=8/18.
We can find the fourth root by taking the square root twice:
taking the square root gives us r</span>²<span> = </span>√<span>8/</span>√<span>18.
Simplifying </span>√<span>8 we get 2</span>√<span>2, and simplifying </span>√<span>18 gives us 3</span>√<span>2; we now have
r</span>²<span>=2</span>√<span>2)/3</span>√<span>2.
The </span>√<span>2 will cancel, leaving r</span>²<span>=2/3. Taking the square root again, we have
r=</span>√<span>2/</span>√<span>3; simplifying this gives us r=</span>√<span>6/3.
We can now work backward to find the sixth term using the seventh one; Divide 8 by </span>√<span>6/3. Dividing by a fraction means multiplying by the reciprocal, so we multiply 8 by 3/</span>√<span>6; this gives us 24/</span>√<span>6, and in a calculator that gives us 9.8 ft.</span>
