the sum of 15 terms of an arithmetic sequence with common difference 6 is 780.write the algebric expression of the sequence
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the sum of 15 terms of an arithmetic sequence with common difference 6 is 780.write the algebric expression of the sequence
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[tex]\huge\sf\purple{\underline{\underline{\pink{Ans}\red{wer:}}}}[/tex]
[tex]\sf{The \ required \ sequence \ is \ 10, \ 16, \ 22,...}[/tex]
[tex]\sf\orange{Given:}[/tex]
[tex]\sf{In, \ an \ AP,}[/tex]
[tex]\sf{\leadsto{Sum \ of \ 15 \ terms(S_{15})=780}}[/tex]
[tex]\sf{\leadsto{Common \ difference (d)=6}}[/tex]
[tex]\sf\pink{To \ find:}[/tex]
[tex]\sf{The \ sequence.}[/tex]
[tex]\sf\green{\underline{\underline{Solution:}}}[/tex]
[tex]\sf{Here, \ d=6, \ S_{15}=780}[/tex]
[tex]\boxed{\sf{S_{n}=\dfrac{n}{2}[2a+(n-1)d]}}[/tex]
[tex]\sf{\therefore{S_{15}=\dfrac{15}{2}[2a+(15-1)\times6]}}[/tex]
[tex]\sf{\therefore{780=\dfrac{15}{2}[2a+14\times6]}}[/tex]
[tex]\sf{\therefore{780=15(a+7\times6)}}[/tex]
[tex]\sf{\therefore{a+42=\dfrac{780}{15}}}[/tex]
[tex]\sf{\therefore{a+42=52}}[/tex]
[tex]\sf{\therefore{a=52-42}}[/tex]
[tex]\sf{\therefore{a=10}}[/tex]
[tex]\sf{The \ required \ sequence \ is}[/tex]
[tex]\sf{t_{1}=a=10,}[/tex]
[tex]\sf{t_{2}=a+d=10+6=16,}[/tex]
[tex]\sf{t_{3}=a+2d=10+2(6)=22.}[/tex]
[tex]\sf\purple{\tt{\therefore{The \ required \ sequence \ is \ 10, \ 16, \ 22,...}}}[/tex]