Math, asked by freefiregamer25, 10 months ago

The sum of the first five terms of an arithmetic sequence is 150 and the sum of the first ten terms is 550
(a) what is the third term of the sequemce​

Answers

Answered by Anonymous
47

 \large\bf\underline{Given:-}

  • sum of first five terms of AP = 150
  • sum of first ten terms of AP = 550

 \large\bf\underline {To \: find:-}

  • 3rd term.

 \huge\bf\underline{Solution:-}

 \bf \: S_n = \frac{n}{2}  [2a + (n - 1)d]

 \rm :  \implies \  \frac{5}{2} [2a +4d  ] = 150 \\  \\   \rm :  \implies \: 5 [2a + 4d] = 300 \\  \\ \rm :  \implies 2a + 4d =  \frac{300}{5}  \\  \\  \rm :  \implies 2a + 4d = 60 ........(i)\\  \\

Sum of first ten terms = 550

 \rm :  \implies  \frac{10}{2} [2a + 9d] = 550 \\  \\  \rm :  \implies5 [2a + 9d] = 550 \\  \\  \rm :  \implies [2a + 9d] =  \frac{550}{5}  \\  \\  \rm :  \implies 2a + 9d = 110.......(ii)

Solving (i) and (ii) by elimination method.

2a +4d = 60

2a +9d = 110

--⠀⠀ -- ⠀⠀--

⠀⠀⠀-5d = -50

d = 50/5

d = 10

putting value of d in (i)

⠀⠀⠀⠀⠀➝ 2a +4×10 = 60

⠀⠀⠀⠀⠀➝ 2a +40 = 60

⠀⠀⠀⠀⠀➝ 2a = 20

⠀⠀⠀⠀⠀➝ a = 20/2

⠀⠀⠀⠀⠀➝ a = 10

So, third term of AP = a +2d

⠀⠀⠀⠀⠀➝ 10 + 2 ×10

⠀⠀⠀⠀⠀➝ 10 + 20

⠀⠀⠀⠀⠀➝ 30

Answered by Anonymous
24

\red{\underline{\underline{Answer:}}}

\sf{The \ third \ term \ of \ the \ A.P. \ is \ 30.}

\sf\orange{Given:}

\sf{In \ an \ A.P.}

\sf{\implies{Sum \ of \ first \ five \ terms \ (S5)=150}}

\sf{\implies{Sum \ of \ first \ ten \ terms \ S(10)=550}}

\sf\pink{To \ find:}

\sf{The \ third \ term \ of \ A.P.}

\sf\green{\underline{\underline{Solution:}}}

\sf{Let \ the \ first \ five \ terms \ of \ A.P. \ be}

\sf{(a-2d), \ (a-d), \ a, \ (a+d) \ and \ (a+2d)}

\sf{According \ to \ the \ first \ condition. }

\sf{(a-2d)+(a-d)+a+(a+d)+(a+2d)=150}

\sf{\therefore{5a=150}}

\sf{\therefore{a=\frac{150}{5}}}

\sf{\therefore{a=30}}

\sf{But, \ t3=a}

\sf{\therefore{t3=30}}

\sf\purple{\tt{\therefore{The \ third \ term \ of \ the \ A.P. \ is \ 30.}}}

Similar questions