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In an AP of the first …

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In an AP of the first term is 2 then the sum of first 5 terms is one-fourth of the sum of the next 5 terms then show that 20th term is -112

Posted by Spandana D 6 years, 6 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 6 months ago

According to question, a=2 and {tex}\mathrm { S } _ { 5 } = \frac { 1 } { 4 } \left[ \mathrm { S } _ { 10 } - \mathrm { S } _ { 5 } \right]{/tex}

{tex}\Rightarrow{/tex} 4S₅ = S₁₀ - S₅

{tex}\Rightarrow{/tex} 5S₅ = S₁₀

{tex}\Rightarrow \sqrt [ 5 ] { \frac { 5 } { 2 } \{ 2 \times 2 + ( 5 - 1 ) d \} } ] = \frac { 10 } { 2 } [ 2 \times 2 + ( 10 - 1 ) d ]{/tex} since {tex}{S_n} = {n \over 2}\left[ {2a + (n - 1)d} \right]{/tex}

{tex}\Rightarrow \frac { 25 } { 2 } [ 4 + 4 d ] = 5 [ 4 + 9 d ]{/tex}

{tex}\Rightarrow{/tex} 25[4 + 4d] = 10[4 + 9d]

{tex}\Rightarrow{/tex} 100 + 100d = 40 + 90d

{tex}\Rightarrow{/tex} 10d = -60

{tex}\Rightarrow{/tex} d = -6

Now, a_n = a + (n - 1)d

{tex}\Rightarrow a _ { 20 } = 2 + ( 20 - 1 ) \times ( - 6 ){/tex}

{tex}\Rightarrow{/tex} a₂₀ = 2 - 114 = -112

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