Question

The sum of first 10 terms of an Arithmetic Progression is 55 and the sum of first 9 terms of the same arithmetic progression is 45. Then its 10 term is​

Answer 1

Answer:

10

Step-by-step explanation:

  For first 10 terms:

⇒ 1st term + 2nd term + 3rd term + ... 10th term

⇒ sum of first 9 term + 10th term

    Given, S₉ = 45, S₁₀ = 55

⇒ S₁₀ = S₉ + T₁₀

⇒ 55 = 45 = T₁₀

⇒ 10 = T₁₀

 Hence the 10th term is 10

                  Technique 2:

S₁₀ = (10/2)[2a + 9d] = 10a + 45d = 55

S₉ = (9/2)[2a + 8d] = 9a + 36d  = 45  

 Solving these equations, we get

a = 1, and d = 1

 Hence 10th term = a + 9d

             = 1 + 9(1) = 10

Answer 2

$\underline{\underline{\textsf{\textbf{\pink{Given\::-}}}}}$

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• Sum of first 10 terms of an Arithmetic Progression is 55.

• Sum of first 9 terms of the same arithmetic progression is 45.

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$\underline{\underline{\textsf{\textbf{\green{To\:Find\::-}}}}}$

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• 10th term of arithmetic progression ?

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$\underline{\underline{\textsf{\textbf{\blue{Solution\::-}}}}}$

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• Let us consider that the terms in arithmetic progression series are : $\bf a_{1},\:a_{2},\:a_{3},\:.\:.\:.,\:a_{10}$

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$\underline{\sf{\bigstar\:According\:to\:the\:Question\::-}}$

$\\ \longrightarrow \:\bf \red{a_{1} + a_{2} + a_{3} +\: .\:.\:.\:+ a_{10} = 55}\:\dashrightarrow \:\purple{\big(eq^{n}\:1\big)}$

$\\ \longrightarrow \:\bf \purple{a_{1} + a_{2} + a_{3} +\: .\:.\:.\:+ a_{9} = 55}\:\dashrightarrow \:\red{\big(eq^{n}\:2\big)}$

$\:$

$\underline{\sf{\bigstar\:Subtracting\:\red{\big(eq^{n}\:2\big)}\:from\:\purple{\big(eq^{n}\:1\big)}\::-}}$

$\\ \longrightarrow \:\sf \big(a_{1} + a_{2} + a_{3} +\: .\:.\:.\:+ a_{10}\big) - \big(a_{1} + a_{2} + a_{3} +\: .\:.\:.\:+ a_{9}\big) = 55 - 45$

$\\ \longrightarrow \:\sf a_{1} + a_{2} + a_{3} +\: .\:.\:.\:+ a_{10} - a_{1} - a_{2} - a_{3} -\: .\:.\:.\:- a_{9} = 10$

$\\ \longrightarrow\: \sf \big(a_{1} - a_{1}\big) + \big(a_{2} - a_{2}\big) + \big(a_{3} - a_{3}\big) +\:.\:.\:.+ \big(a_{9} - a_{9}\big) + a_{10} = 10$

$\\ \longrightarrow \:\boxed{\bf{\green{a_{10} = 10}}}\:\pink{\bigstar}$

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$\underline{\sf{\bigstar\:Direct\:method\::-}}$

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We know that ::

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$\quad\odot\:{\underline{\boxed{\bf{\blue{T_{n} = S_{n} - S_{\big(n-1\big)} }}}}}$

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Putting all known values ::

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$\\ \longrightarrow\: \sf T_{10} = S_{10} - S_{\big(10 - 1\big)}$

$\\ \longrightarrow\: \sf T_{10} = S_{10} - S_{9}$

$\\ \longrightarrow\: \sf T_{10} = 55 - 45$

$\\ \longrightarrow \:\boxed{\bf{\green{T_{10} = 10}}}\:\pink{\bigstar}$

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$\therefore\:{\underline{\sf{Hence,\:10^{th}\:term\:of\:arithmetic\:progression\:is\:{\textit{\textbf{10}}}}}}$

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$\underline{\underline{\textsf{\textbf{\red{Learn\:more\:on\:brainly\::-}}}}}$

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$\underline{\sf{\bigstar\:Question\::-}}$

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Divide 56 in four parts in AP such that the ratio of the product of their extremes

(1st and 4th) to the product of means (2nd and 3rd) is 5:6.

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$\underline{\sf{\bigstar\:Answer\::-}}$

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https://brainly.in/question/42294457

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