AA SL · Sequences · Logarithms
An arithmetic sequence has first term 
and common difference 
.
The 13th term of the sequence is 
.
Find the value of 
.
Hint
Use the formula for the 
th term of an arithmetic sequence:
![\[ u_n=u_1+(n-1)d. \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-0e159638622a0257f83ca4e9be77f8cd_l3.png "Rendered by QuickLaTeX.com")
You will also need to simplify 
.
Solution
Using
![\[ u_n=u_1+(n-1)d, \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-9e40192b1d1050a19e57f400c7a44cac_l3.png "Rendered by QuickLaTeX.com")
we have
![\[ \ln a+12\ln 3=8\ln 9. \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-cf583d581ed195455841b37a9f61108d_l3.png "Rendered by QuickLaTeX.com")
Since
![\[ \ln 9=\ln(3^2)=2\ln 3, \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-9a860a0ad1959b92003f9e337f6496f6_l3.png "Rendered by QuickLaTeX.com")
it follows that
![\[ \ln a+12\ln 3=16\ln 3. \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-b30810d3d97e94bbc816610f0a6d5949_l3.png "Rendered by QuickLaTeX.com")
Hence
![\[ \ln a=4\ln 3=\ln(3^4). \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-6aacd3fad099a9b9d9939ae9f3b9b386_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[ \boxed{a=81}. \]](https://www.ibmathematics.org/wp-content/ql-cache/quicklatex.com-6b0660418471d2e3937fda8442665bcb_l3.png "Rendered by QuickLaTeX.com")
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