Question

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

Answer 1

Answer:

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Answer 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

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

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