Question

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Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the concept of Arithmetic Progression had been used. We see we are given the sum nₜₕ term. From this we can derive a equation to find the nₜₕ term by using formula of sum of x terms. After finding the value of n, we can apply values and find the answer that is x.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{S_{n}\;=\;\bf{\dfrac{n}{2}\;[2a\;+\;(n\;-\;1)d]}}}$

$\\\;\boxed{\sf{a_{n}\;=\;\bf{a\;+\;(n\;-\;1)d}}}$

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★ Solution :-

Given,

» First term of the A.P. = a = 1

» Second term of the A.P. = a₂ = 4

» Third term of the A.P. = a₃ = 7

» Common Difference, d = a₂ - a = 4 - 1 = 3

» nₜₕ term = x

» Sₙ = 287

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~ For the value of n ::

We know that,

$\\\;\;\displaystyle{\sf{:\rightarrow\;\;S_{n}\;=\;\bf{\dfrac{n}{2}\;[2a\;+\;(n\;-\;1)d]}}}$

By applying the values here, we get,

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\dfrac{n}{2}\;[2(1)\;+\;(n\;-\;1)(3)]\;=\;\bf{287}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\dfrac{n}{2}\;[2\;+\;3n\;-\;3]\;=\;\bf{287}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\dfrac{n}{2}\;[3n\;-\;1]\;=\;\bf{287}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;n\;[3n\;-\;1]\;=\;\bf{287\;\times\;2}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;3n^{2}\;-\;n\;=\;\bf{574}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;3n^{2}\;-\;n\;-\;574\;=\;\bf{0}}}$

This forms a Quadratic Equation. Using the method of Splitting the Middle Term, we get,

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;3n(n\;-\;14)\;-\;41(n\;-\;14)\;=\;\bf{0}}}$

Since, 574 = 14 × 41

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;(3n\;-\;41)(n\;-\;14)\;=\;\bf{0}}}$

Here,

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\tt{either}\;\;\sf{(3n\;-\;41)}\;=\;\bf{0}\;\quad\;\tt{or}\;\;\sf{(n\;-\;14)}\;=\;\bf{0}}}$

Then,

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\sf{3n}\;=\;\bf{41}\;\quad\;\tt{or}\;\;\sf{n}\;=\;\bf{14}}}$

$\\\;\;\displaystyle{\sf{:\Rightarrow\;\;\sf{n}\;=\;\bf{\dfrac{41}{3}}\;\quad\;\tt{or}\;\;\sf{n}\;=\;\bf{14}}}$

Here we got two values of n. We know that, n cannot be negative since its the number of term in sequence. Then,

$\\\;\;\displaystyle{\underline{\bf{:\Rightarrow\;\;n\;=\;\bf{\green{14}}}}}$

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~ For the value of x ::

We know that, nₜₕ term is x. And we have the required values to find the nₜₕ term. This means 14th term is x.

Then,

$\\\;\displaystyle{\sf{:\mapsto\;\;a_{n}\;=\;\bf{a\;+\;(n\;-\;1)d}}}$

$\\\;\displaystyle{\sf{:\mapsto\;\;x\;=\;\bf{1\;+\;(14\;-\;1)(3)}}}$

$\\\;\displaystyle{\sf{:\mapsto\;\;x\;=\;\bf{1\;+\;(13)(3)}}}$

$\\\;\displaystyle{\sf{:\mapsto\;\;x\;=\;\bf{1\;+\;39}}}$

$\\\;\displaystyle{\underline{\bf{:\mapsto\;\;x\;=\;\bf{\purple{40}}}}}$

$\\\;\underline{\boxed{\tt{The\;\;value\;\;of\;\;x\;=\;\bf{\blue{40}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;S_{n}\;=\;\dfrac{n}{2}\:[n\;+\;1]}$

$\\\;\sf{\leadsto\;\;l\;=\;a\;+\;(n\;-\;1)(-d)}$

$\\\;\sf{\leadsto\;\;Arithmetic\;Mean\;=\;\dfrac{a\;+\;b}{2}}$

$\\\;\sf{\leadsto\;\;S_{n}\;=\;\dfrac{n}{2}\:[a_{n}\;+\;a]}$

Answer 2

Answer:

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