Question

In an AP, the sum of first five terms is 1/4 the sum of the next five terms.Then show that the 20th term is -112​

Answer 1

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♠ In an AP, the sum of first five terms is 1/4 the sum of the next five terms.Then show that the 20th term is -112.

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${\color{maroon}{\textsf{\textbf{Giνєη}}}}$

First term, a → 2

Sum of first 5 terms = ¼[ Sum of next five terms ]

Using the formula for Sum of n terms :

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

For Sum of first five terms ,

$\sf \implies \: S_5 = \frac{5}{2} [ \: 2 \times 2 + (5 - 1)d \: ] \\ \\\sf \implies \: S_5 = \frac{5}{2} \: \: [ 4 + 4d] \\ \\ \sf \implies \: S_5 = \frac{5}{2} \times 4 \: \: [ 1+ d] \\ \\ { \bold{ \sf \: S_5 = 10[1 + d]}}$

For the Sum of first 10 terms ,

$\sf \implies S 10 = \frac{10}{2} [ \: 2 \times 2 + (10 - 1)d \: ] \\ \\\sf \implies { \bold { \sf \: S10 = 5 [ 4 + 9d \ ]}}$

Now , as per question :

$\sf \implies S_5 = \frac{1}{4} \: [ \: S10 - \: S_5 \: ] \\ \\\sf \implies 10 + 10d = \frac{1}{4} \: [ \: 20 + 45d \: - 10 - 10d \: ] \\ \\ \sf \implies40 + 40d = 10 + 35d \\ \\ \sf \implies 5d = - 30 \\ \\\sf \implies { \boxed{ \bold { \sf \red{d = - 6}}}}$

•°• Sum of first 20 terms :-

$\sf \implies \: S20 = \frac{20}{2} \: [ \: 2 \times 2 + (20 - 1) - 6 \: ] \\ \\ \sf \implies \: S20 = 10 \: (4 + 19 \times ( - 6))$

$•°• {\underline{\bold {S20 = -1100}}}$