Math, asked by shantabellatimath123, 10 months ago

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The sum of the 4th and 10th terms of an A.P. is 10 and their product is 24. Find the sum of the first 25 terms of
the A.P​

Answers

Answered by mddilshad11ab
91

\sf\large\underline{Given:}

  • \rm{The\:sum\:of\:4th\:10th\: terms=10}
  • \rm{The\: product\:of\: their\:is 24}

\sf\large\underline{To\: Find:}

  • \rm{The\:sum\:of\:1st\:25th\:terms\:of\:AP}

\sf\large\underline{Let:}

  • \rm{The\:1st\:term\:of\:AP=a}
  • \rm{The\: common\: difference=d}

\sf\large\underline{Solution:}

\rm{\implies 4th+10th\: terms=10}

\rm{\implies a+3d+a+9d=10}

\rm{\implies 2a+12d=10}

  • Dividing by 2 on both sides]

\rm{\implies a+6d=5}

\rm\green{\implies a=5-6d-----(i)}

\rm{\implies 4th\times10th\: terms=24}

\rm{\implies (a+3d)(a+9d)=24}

\rm{\implies a^2+12da+27d^2=24}

\rm\green{\implies a^2+12da+27d^2-24=0------(ii)}

  • Putting the value of a=5-6d in eq ii ]

\rm{\implies a^2+12da+27d^2-24=0}

\rm{\implies (5-6d)^2+12d(5-6d)+27d^2-24=0}

\rm{\implies 25-60d+36d^2+60d-72d^2+27d^2-24=0}

\rm{\implies -9d^2+1=0}

\rm{\implies -9d^2=-1}

\rm{\implies d^2=\dfrac{-1}{-9}}

\rm{\implies d=\dfrac{1}{3}}

  • Putting the value of d in eq I]

\rm{\implies a=5-6d}

\rm{\implies a=5-6\times\dfrac{1}{3}}

\rm{\implies a=5-2\times1}

\rm{\implies a=5-2}

\rm\red{\implies a=3}

  • Now, calculate 1st 25th terms here]

\rm\purple{\implies S_n=\dfrac{n}{2}[2a+(n-1)d]}

\rm{\implies S_{25}=\dfrac{25}{2}[2\times3+(25-1)\times\dfrac{1}{3}]}

\rm{\implies S_{25}=\dfrac{25}{2}[6+24\times\dfrac{1}{3}]}

\rm{\implies S_{25}=\dfrac{25}{2}[6+8]}

\rm{\implies S_{25}=\dfrac{25}{2}\times14}

\rm{\implies S_{25}=25\times7}

\rm\purple{\implies S_{25}=175}

Hence,

  • The sum of 1st 25th term's is 175
Answered by BendingReality
32

Answer:

S₂₅ = 175 and  75

For [ a = 3 and d = 1 / 3 ] and [ a = 7 and d = - 1 / 3 ] respectively!

Step-by-step explanation:

Given :

t₄ + t₁₀ = 10

= > a + 3 d + a + 9 d = 10

= > 2 a + 12 d = 10

= > a = 5 - 6 d .... ( i )

Also given :

t₄ . t₁₀ = 24

= > ( a + 3 d ) ( a + 9 d ) = 24

Putting values of a = 5 - 6 d from ( i )

= > ( 5 - 3 d ) ( 5 + 3 d ) = 24

= > 5² - 9 d² = 24

= > d² = 1 / 9

= > d = ± 1 / 3

Now putting in first :

= > a = 5 - 6 × 1 / 3

= > a = 3 OR a = 7

Sum of first 25 terms as :

S₂₅ = 25 / 2 ( 6 + 24 / 3 )  and   25 / 2 ( 14 - 24 / 3 )

= > S₂₅ = 25 ( 7 )  and   25 / 2 ( 6 )

= > S₂₅ = 175 and  75

Hence we get required answer!

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