Math, asked by neethummol, 2 months ago

in an AP if the first term is 38 and 16th term is 73 then find its 31st term​

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Answered by sonakshi605
0

Answer:

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Answered by Anonymous
2

Given, that  \sf \:{ a_{11} = 38} \: and \: {a_{16}} = 73

If the first term a and the common difference d

We know that,  \bf \red {a_n = a + (n - 1)d} , Using this formula to find  \red{ \bf{ {n}^{th} }}Term of arithmetic progression

 \: \sf \: given \:  {11}^{th}  \: term \: is \: 38 \\  \\ \sf\twoheadrightarrow a_{11}  = a + (11 - 1)d \\  \\ \sf\twoheadrightarrow38 = a + (11 -1 )d \\  \\ \sf\twoheadrightarrow38 = a +1 0d \\  \\ \sf\twoheadrightarrow38 - 10d = a \\  \\  \green{\sf\twoheadrightarrow \: a = 38 - 10d \:  \:  \:  \:  \:  -  -  -  -  - (1)}

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\sf \: given \:  {16}^{th}  \: term \: is \: 73 \\  \\ \sf\twoheadrightarrow \: a_{16}  =  a + (16 - 1)d \\  \\  \sf\twoheadrightarrow73 = a + (16 - 1) \\  \\  \sf\twoheadrightarrow73 = a + 15d \\  \\ \sf\twoheadrightarrow73 - 15d = a \\  \\  \red{\sf\twoheadrightarrow \: a = 73 - 15d \:  \:  \:  \:  -  -  -  -  - (2)}

 \qquad \_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_\_ \_

Subtracting equation (1) from equation (2) we get following:

\sf\longmapsto38 - 11d = 73 - 15d \\  \\ \sf\longmapsto38 - 73 =   15d - 10d \\  \\ \sf\longmapsto - 35 =  5d \\  \\ \sf\longmapsto \frac{ \cancel-  \cancel{35}}{  \cancel-  \cancel5}   \:  \:  \: \\  \\\sf\longmapsto7 = d  \\  \\  \purple{\sf\longmapsto \: d = 7}

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 \sf \: substituting \: the \: value \: of \: \bf{d} \sf \: in \: equation \: (1)  \\  \\ \sf\longmapsto a = 38 - 10d \\  \\ \sf\longmapsto \: a = 38 - 10 \times 7 \\  \\ \sf\longmapsto \: a = 38 - 70  \\  \\ \sf\longmapsto \: a =  - 32 \\  \\    \blue{\boxed{\sf\longmapsto \: a =  - 32}}

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Now 31st term can be calculated as follows:

 \tt So, n = 31, a = - 32, d = 7

 \sf \: We  \: know  \: that \:  \\  \boxed{ \blue{ a_n = a + (n+1)d}}

Putting values in formals

\sf\longrightarrow \: a_{n} = a + (n + 1)d \\  \\ \sf\longrightarrow \: a_{31} =  - 32 + (31 - 1)7 \\  \\  \sf=  - 32 + 30 \times 7 \\  \\  =  \sf \:  - 32 + 210 \\  \\  \sf \:  = 178

 \sf \: there four \: the \:  {31}^{st}  \: term \: of \: arithmetic \: progression \:\orange{178}

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