In an ap 21 terms the sum of first 3term is -33 and that of middle 3 i

1.	In an ap 21 terms the sum of first 3term is -33 and that of middle 3 is 75 what is the sum of the ap ?
Answer» $\blue{\bold{\underline{\underline{Answer:}}}}$ $\green{\tt{\therefore{Sum\:of\:A.P=2037}}}$ $\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$ $\green{\underline \bold{<klux>GIVEN</klux> :}} \\ \tt: \implies Number \: of \: terms = 21 \\ \\ \tt: \implies Sum \: of \: first \: 3 \: terms = - 33 \\ \\ \tt: \implies Sum \: of \: next \: three = 75 \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Sum \:of \: A.P = ?$ • ACCORDING to given QUESTION : $\bold{For \: sum \: of \: first \: three \: term} \\ \tt: \implies s_{3} = \frac{3}{2}(2a + (3 - 1) \times d) \\ \\ \tt: \implies - 33 = \frac{3}{2} (2a + 2d) \\ \\ \tt: \implies - 33 = 3(a + d) \\ \\ \tt: \implies a + d = - 11 - - - - - (1) \\ \\ \bold{For \: next \: three \: sum \: of \: terms} \\ \tt: \implies 75 = \frac{3}{2} (2(a + 3d) + (3 - 1)d) \\ \\ \tt: \implies \frac{150}{3} = 2a + 6d + 2d \\ \\ \tt: \implies 50 = 2a + 8d \\ \\ \tt: \implies a + 4d = 25 - - - - - (2) \\ \\ \text{Subtracting \: (1) \: from \: (2)} \\ \tt: \implies 4d - d = 25 - ( - 11)$ $\tt: \implies 3d = 36 \\ \\ \tt: \implies d = \frac{36}{3} \\ \\ \green{\tt: \implies d = 12} \\ \\ \text{Putting \: value \: of \: d \: in \: (2)} \\ \tt: \implies a + 4 \times 12 = 25 \\ \\ \tt: \implies a + 48 = 25 \\ \\ \tt: \implies a = 25 - 48 \\ \\ \green{\tt: \implies a = - 23} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies s_{n} = \frac{n}{2} (2a + (n - 1)d) \\ \\ \tt: \implies s_{21} = \frac{21}{2} (2 \times - 23 + (21 - 1) \times 12) \\ \\ \tt: \implies s_{21} = \frac{21}{2} ( - 46 + 240) \\ \\ \tt: \implies s_{21} = \frac{21}{2} \times 194 \\ \\ \green{\tt: \implies s_{21} =2037 }$

In an ap 21 terms the sum of first 3term is -33 and that of middle 3 is 75 what is the sum of the ap ?

Answer»

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Sum\:of\:A.P=2037}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{<klux>GIVEN</klux> :}} \\ \tt: \implies Number \: of \: terms = 21 \\ \\ \tt: \implies Sum \: of \: first \: 3 \: terms = - 33 \\ \\ \tt: \implies Sum \: of \: next \: three = 75 \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Sum \:of \: A.P = ?$

• ACCORDING to given QUESTION :

$\bold{For \: sum \: of \: first \: three \: term} \\ \tt: \implies s_{3} = \frac{3}{2}(2a + (3 - 1) \times d) \\ \\ \tt: \implies - 33 = \frac{3}{2} (2a + 2d) \\ \\ \tt: \implies - 33 = 3(a + d) \\ \\ \tt: \implies a + d = - 11 - - - - - (1) \\ \\ \bold{For \: next \: three \: sum \: of \: terms} \\ \tt: \implies 75 = \frac{3}{2} (2(a + 3d) + (3 - 1)d) \\ \\ \tt: \implies \frac{150}{3} = 2a + 6d + 2d \\ \\ \tt: \implies 50 = 2a + 8d \\ \\ \tt: \implies a + 4d = 25 - - - - - (2) \\ \\ \text{Subtracting \: (1) \: from \: (2)} \\ \tt: \implies 4d - d = 25 - ( - 11)$

$\tt: \implies 3d = 36 \\ \\ \tt: \implies d = \frac{36}{3} \\ \\ \green{\tt: \implies d = 12} \\ \\ \text{Putting \: value \: of \: d \: in \: (2)} \\ \tt: \implies a + 4 \times 12 = 25 \\ \\ \tt: \implies a + 48 = 25 \\ \\ \tt: \implies a = 25 - 48 \\ \\ \green{\tt: \implies a = - 23} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies s_{n} = \frac{n}{2} (2a + (n - 1)d) \\ \\ \tt: \implies s_{21} = \frac{21}{2} (2 \times - 23 + (21 - 1) \times 12) \\ \\ \tt: \implies s_{21} = \frac{21}{2} ( - 46 + 240) \\ \\ \tt: \implies s_{21} = \frac{21}{2} \times 194 \\ \\ \green{\tt: \implies s_{21} =2037 }$

In an ap 21 terms the sum of first 3term is -33 and that of middle 3 is 75 what is the sum of the ap ?

• ACCORDING to given QUESTION :

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