Notes: ``3rd term:` a_1 = 16, a_4 = 27/4` Find the indicated term of the geometric sequence.

Saturday, May 16, 2009

3rd term: $\displaystyle{a}_{{1}}={16},{a}_{{4}}=\frac{{27}}{{4}}$ Find the indicated term of the geometric sequence.

You need to write the following formula such that:

a_4 = a_3*q

$\displaystyle{a}_{{3}}={a}_{{2}}\cdot{q}$

$\displaystyle{a}_{{2}}={a}_{{1}}\cdot{q}$

Replace $\displaystyle{a}_{{1}}\cdot{q}$ for $\displaystyle{a}_{{2}}$ , such that:

$\displaystyle{a}_{{3}}={a}_{{1}}\cdot{q}\cdot{q}\Rightarrow{a}_{{3}}={a}_{{1}}\cdot{{q}}^{{2}}$

Replace $\displaystyle{a}_{{1}}\cdot{{q}}^{{2}}$ for $\displaystyle{a}_{{3}}$ such that:

$\displaystyle{a}_{{4}}={a}_{{1}}\cdot{{q}}^{{2}}\cdot{q}\Rightarrow{a}_{{4}}={a}_{{1}}\cdot{{q}}^{{3}}\Rightarrow{{q}}^{{3}}=\frac{{{a}_{{4}}}}{{{a}_{{1}}}}\Rightarrow{q}={\sqrt[{{3}}]{{\frac{{{a}_{{4}}}}{{{a}_{{1}}}}}}}$

$\displaystyle{q}={\sqrt[{{3}}]{{\frac{{\frac{{27}}{{4}}}}{{{16}}}}}}\Rightarrow{q}={\sqrt[{{3}}]{{\frac{{{{3}}^{{3}}}}{{{{4}}^{{3}}}}}}}$

$\displaystyle{q}=\frac{{3}}{{4}}$

You may find $\displaystyle{a}_{{3}}$ such that:

$\displaystyle{a}_{{3}}=\frac{{{a}_{{4}}}}{{q}}\Rightarrow{a}_{{3}}=\frac{{\frac{{27}}{{4}}}}{{\frac{{3}}{{4}}}}\Rightarrow{a}_{{3}}={9}$

Hence, evaluating $\displaystyle{a}_{{3}}$ yields $\displaystyle{a}_{{3}}={9}.$

Notes

Saturday, May 16, 2009

3rd term: $\displaystyle{a}_{{1}}={16},{a}_{{4}}=\frac{{27}}{{4}}$ Find the indicated term of the geometric sequence.

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How does the choice of details set the tone of the sermon?

Saturday, May 16, 2009

3rd term: Find the indicated term of the geometric sequence.

No comments:

Post a Comment

How does the choice of details set the tone of the sermon?

3rd term: $\displaystyle{a}_{{1}}={16},{a}_{{4}}=\frac{{27}}{{4}}$ Find the indicated term of the geometric sequence.