Question

Which of the following sequences are A.P. ? If they are A.P. find the common
difference of
2, 5/2, 3, 7/3,....​

Answer 1

Given terms,

• $\sf a_1 = 2, \: a_2 = \dfrac{5}{2}, \; a_3 = 3, \: a_4 = \dfrac{7}{3}$

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$\dag\;{\underline{\frak{As \ We \ know \ that,}}}$⠀

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$\star\:\boxed{\sf{\pink{d_{\:(common\: difference)} = a_2 - a_1}}}$

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Therefore,

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$:\implies\sf d_1 = \dfrac{5}{2} - 2 \\\\\\:\implies{\underline{\boxed{\frak{\purple{ d_1 = \dfrac{1}{2}}}}}}\:\bigstar$

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$:\implies\sf d_2 = 3 - \dfrac{5}{2} \\\\\\:\implies{\underline{\boxed{\frak{\purple{d_2 = \dfrac{1}{2}}}}}}\:\bigstar$

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$:\implies\sf d_3 = \dfrac{7}{3} - 3\\\\\\:\implies{\underline{\boxed{\frak{\purple{d_3 = \dfrac{-1}{\:3}}}}}}\:\bigstar$

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Here, we can see that $\sf d_1 = d_2 \neq d_3$ So, the given terms aren't in AP(Arithmetic Progression).

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$\qquad\qquad{\underline{\underline{\dag \: \bf\:Formulaes \ of \: the \:AP\: :}}}$

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• To find nth term of the AP(Arithmetic Progression) formula is given as, $\sf a_n = a+ (n - 1)d.$

• To find out the Sum of the AP = $\sf S_n = \dfrac{n}{2} \bigg( 2a + (n - 1)d \bigg).$⠀

• To find out the sum of all terms have the last term of the AP 'l' = $\sf \dfrac{n}{2}(a + l).$

Answer 2

Answer:

Required Answer :-

As we know that

$\sf \: D_{common} = a_2 - a_1$

$\sf \: D = \dfrac{5}{2} - 2$

$\sf \: D = \dfrac{5 - 4}{2} = \dfrac{1}{2}$

$\sf \: D_2 = 3 - \dfrac{5}{2}$

$\sf \: D_2 = \dfrac{6 - 5}{2} = \dfrac{1}{2}$

$\sf \: D_3 = \dfrac{7}{3} - 3$

$\sf \: D_3 = \dfrac{7 - 9}{3}$

$\sf \: D_3 = \dfrac{ - 2}{3}$

Hence :-

We can say that they aren't in AP