The first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the seq

Question

The first term of a sequence along with a recursion formula for the remaining terms is given below. Write out the first ten terms of the sequence.

a1=6,an+1=an+(1/3^n)

simplify your answer

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Mia 1 week 2021-09-13T11:31:44+00:00 1 Answer 0

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    2021-09-13T11:33:05+00:00

    Answer:

    See below

    Step-by-step explanation:

    Let s first find a simplified expression for the sum of the first n terms of the geometric sequence \bf \left\{\displaystyle\frac{1}{3^k}\right\}_{k=1}^n

    \bf S_n=\displaystyle\frac{1}{3}+\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{3^3}+...+\displaystyle\frac{1}{3^n}

    If we multiply both sides by 1/3, we get

    \bf \displaystyle\frac{1}{3}S_n=\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{3^3}+...+\displaystyle\frac{1}{3^n}+\displaystyle\frac{1}{3^{n+1}}

    Hence

    \bf S_n-\displaystyle\frac{1}{3}S_n=\displaystyle\frac{1}{3}-\displaystyle\frac{1}{3^{n+1}}\Rightarrow \displaystyle\frac{2}{3}S_n=\displaystyle\frac{1}{3}-\displaystyle\frac{1}{3^{n+1}}\Rightarrow\\\\\Rightarrow 2S_n=1-\displaystyle\frac{1}{3^n}=\displaystyle\frac{3^n-1}{3^n}\Rightarrow S_n=\displaystyle\frac{3^n-1}{2(3^n)}

    Now, we have

    \bf a_1=6\\\\a_2=6+\displaystyle\frac{1}{3}=\displaystyle\frac{19}{3}\\\\a_3=6+\displaystyle\frac{1}{3}+\displaystyle\frac{1}{3^2}=6+S_2=6+\displaystyle\frac{8}{18}=\displaystyle\frac{58}{9}\\\\a_4=6+S_3=6+\displaystyle\frac{26}{54}=\displaystyle\frac{175}{27}

    \bf a_5=6+S_4=6+\displaystyle\frac{3^4-1}{2(3^4)}=\displaystyle\frac{526}{81}\\\\a_6=6+S_5=6+\displaystyle\frac{3^5-1}{2(3^5)}=\displaystyle\frac{1579}{243}\\\\a_7=6+S_6=6+\displaystyle\frac{3^6-1}{2(3^6)}=\displaystyle\frac{4738}{729}

    \bf a_8=6+S_7=6+\displaystyle\frac{3^7-1}{2(3^7)}=\displaystyle\frac{14215}{2187}\\\\a_9=6+S_8=6+\displaystyle\frac{3^8-1}{2(3^8)}=\displaystyle\frac{42646}{6561}\\\\a_{10}=6+S_9=6+\displaystyle\frac{3^9-1}{2(3^9)}=\displaystyle\frac{127939}{19683}

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27:3+15-4x7+3-1=? ( )