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Help me please !!

The sum of 5th and 9th terms of an A.P. is 72 and the sum of 7th and 12th terms is 97. Find the A.P.

Respuesta :

ItzBrainlyCommando

[tex] \star\:{\underline{\underline{\sf{\purple{Given ::}}}}}[/tex]

❖ [tex]\sf a_{5} + a_{9} = 72 [/tex]

❖ [tex]\sf a_{7} + a_{12}= 97 [/tex]

[tex] \star\:{\underline{\underline{\sf{\purple{To \: Find ::}}}}}[/tex]

❖ The A.P

[tex] \star\:{\underline{\underline{\sf{\purple{Solution ::}}}}}[/tex]

Let,

[tex]a[/tex] be the first term and [tex]d[/tex] be the common difference of the A.P

According to the question,

[tex]\sf a_{5} + a_{9} = 72 [/tex] and [tex]\sf a_{7} + a_{12}= 97 [/tex]

[tex]\longrightarrow \sf (a + 4d) + (a + 8d) = 72[/tex] and [tex]\sf (a + 6d) + (a + 11d) = 97[/tex]

Thus, we have

[tex] \longrightarrow \sf 2a + 12d = 72 - - (i)[/tex]

[tex] \longrightarrow \sf 2a + 17d = 97 - - (ii) [/tex]

Subtracting (i) from (ii), we get

[tex] \implies \sf 5d = 25[/tex]

[tex] \implies \sf d = \dfrac{25}{5} [/tex]

[tex]\implies {\star{ \underline{\boxed{\sf{\pink{\sf d = 5}}}}}}[/tex]

Now,

Putting d=5 in (i), we get

[tex] \longrightarrow \sf 2a + 12(5) = 72 [/tex]

[tex] \longrightarrow \sf 2a + 60= 72 [/tex]

[tex]\longrightarrow \sf 2a= 72 - 60[/tex]

[tex]\longrightarrow \sf 2a= 12[/tex]

[tex]\longrightarrow \sf a= \dfrac{12}{2} [/tex]

[tex]\longrightarrow{\star{ \underline{\boxed{\sf{\pink{\sf a = 6}}}}}}[/tex]

[tex] \therefore[/tex] a=6 and d=5

Hence, the A.P is 6,11,16,21,26...

[tex]\rule{250pt}{2.5pt}[/tex]

TwinklingStar

Given:-

  • The sum of the 5th and 9th terms of an A.P is 72 and the sum of 7th and 12th terms is 97.

To find:-

  • we have to find the A.P. = ?

Solution:-

Let, the first term of A.P be 'a' and common difference is 'd'...!

[tex] \\ \maltese \: \: {\underline{ \sf {The \: \: sum \: \: of \: \: a_5 \: \: and \: \: a_9 \: \: is \: \: 72 }}} \\ [/tex]

[tex] \\ \sf\implies A_5 \: + \: A_9=72 \: \\ [/tex]

[tex] \\ \sf\implies a + 4_d + a + 8_d = 72 \\ [/tex]

[tex] \\ \sf\implies 2_a+12_d=72 \qquad \qquad - (i) \\ \\ [/tex]

[tex] \\ \maltese \: \: {\underline{ \sf {The \: \: sum \: \: of \: \: a_7 \: \: and \: \: a_{12} \: \: is \: \: 97 }}} \\ [/tex]

[tex] \\ \sf\implies A_7 \: + \: A_{12}=97 \: \\ [/tex]

[tex] \\ \sf\implies a + 6_d + a + 11_d = 97 \\ [/tex]

[tex] \\ \sf\implies 2_a+17_d=97 \qquad \qquad - (ii) \\ \\ [/tex]

[tex] \\ \maltese \: \: {\underline{ \sf {Subtract \: \: equation \: (i) \: from \: \: equation \: (ii), \: we \: \: get }}} \\ [/tex]

[tex] \\ \sf \implies \: 2_a + 17_d = 97 \\ [/tex]

[tex] \\ \sf \implies \: - (2_a + 12_d) = 72 \\ [/tex]

[tex] \\ \sf \implies \: 5_d = 25 \\ [/tex]

[tex] \\ \implies{\underline{\boxed{\sf {d=5 }}}} \\\\ [/tex]

[tex] \\ \maltese \: \: {\underline{ \sf {Put \: \: the \: \: value \: \: of \: \: d \: \: in \: \: Equation \: \: (i) }}} \\ [/tex]

[tex] \\ \sf\implies 2_a + 12_d = 72 \\ [/tex]

[tex] \\ \sf\implies 2_a+12_{(5)}=72 \\ [/tex]

[tex] \\ \sf\implies 2_a + 60 = 72 \\ [/tex]

[tex] \\ \sf\implies 2_a=72-60 \\ [/tex]

[tex] \\ \sf\implies 2_a = 12 \\ [/tex]

[tex] \\ \implies\sf a = \frac{12}{2} \\ [/tex]

[tex] \\ \implies{\underline{\boxed{\sf {a=6 }}}} \\ [/tex]

Henceforth the first term of A.P is 'a = 6' and common difference is 'd= 5'...!

∴ A.P = 6 , 11 , 16 , 21.....

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