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Find the value of a_2 + a_4 + ... + a_{98} if a_1, a_2, a_3, ... is an arithmetic progression with common difference 1 and a_1 + a_2 + ... + a_{98} = 99.

 Jun 14, 2021

Best Answer 

 #1
avatar+26162 
+2

Find the value of \(a_2 + a_4 + \ldots + a_{98}\)

if \(a_1, a_2, a_3, \ldots\) is an arithmetic progression with common difference 1

and \(a_1 + a_2 + ... + a_{98} = 99\).

 

\(\begin{array}{|rcll|} \hline a_1 + a_2 + ... + a_{98} &=& 99 \\\\ \left(\dfrac{a_1 + a_{98}}{2}\right)*98 &=& 99 \\ \mathbf{a_1 + a_{98}} &=& \mathbf{\dfrac{ 2*99 }{98} } \\ \hline \end{array}\)

 

\(\text{Let $x=a_2 + a_4 + \ldots + a_{98}$}\\ \begin{array}{|rcll|} \hline x &=& a_2 + a_4 + \ldots + a_{98} \\\\\ x &=& \left(\dfrac{a_2 + a_{98}}{2}\right)*\dfrac{98}{2} \quad | \quad a_2 = a_1 +1 \\\\ x &=& \left(\dfrac{1 + a_1 + a_{98}}{2}\right)*\dfrac{98}{2} \\\\ x &=& (1 + a_1 + a_{98})*\dfrac{98}{4} \\\\ x &=& \dfrac{98}{4} + (a_1 + a_{98})*\dfrac{98}{4} \quad | \quad \mathbf{a_1 + a_{98}=\dfrac{ 2*99 }{98} } \\\\ x &=& \dfrac{98}{4} + \dfrac{ 2*99 }{98}*\dfrac{98}{4} \\\\ x &=& \dfrac{98}{4} + \dfrac{ 2*99 }{4} \\\\ x &=& \dfrac{98+2*99}{4} \\\\ \mathbf{x} &=& \mathbf{74} \\ \hline \end{array}\)

 

\(a_2 + a_4 + \ldots + a_{98} = \mathbf{74} \)

 

laugh

 Jun 14, 2021
 #1
avatar+26162 
+2
Best Answer

Find the value of \(a_2 + a_4 + \ldots + a_{98}\)

if \(a_1, a_2, a_3, \ldots\) is an arithmetic progression with common difference 1

and \(a_1 + a_2 + ... + a_{98} = 99\).

 

\(\begin{array}{|rcll|} \hline a_1 + a_2 + ... + a_{98} &=& 99 \\\\ \left(\dfrac{a_1 + a_{98}}{2}\right)*98 &=& 99 \\ \mathbf{a_1 + a_{98}} &=& \mathbf{\dfrac{ 2*99 }{98} } \\ \hline \end{array}\)

 

\(\text{Let $x=a_2 + a_4 + \ldots + a_{98}$}\\ \begin{array}{|rcll|} \hline x &=& a_2 + a_4 + \ldots + a_{98} \\\\\ x &=& \left(\dfrac{a_2 + a_{98}}{2}\right)*\dfrac{98}{2} \quad | \quad a_2 = a_1 +1 \\\\ x &=& \left(\dfrac{1 + a_1 + a_{98}}{2}\right)*\dfrac{98}{2} \\\\ x &=& (1 + a_1 + a_{98})*\dfrac{98}{4} \\\\ x &=& \dfrac{98}{4} + (a_1 + a_{98})*\dfrac{98}{4} \quad | \quad \mathbf{a_1 + a_{98}=\dfrac{ 2*99 }{98} } \\\\ x &=& \dfrac{98}{4} + \dfrac{ 2*99 }{98}*\dfrac{98}{4} \\\\ x &=& \dfrac{98}{4} + \dfrac{ 2*99 }{4} \\\\ x &=& \dfrac{98+2*99}{4} \\\\ \mathbf{x} &=& \mathbf{74} \\ \hline \end{array}\)

 

\(a_2 + a_4 + \ldots + a_{98} = \mathbf{74} \)

 

laugh

heureka Jun 14, 2021
 #2
avatar+121006 
+1

Very nice , heureka   !!!!!

 

 

cool cool cool

CPhill  Jun 14, 2021
 #3
avatar+26162 
+1

Thank you, CPhill !

 

laugh

heureka  Jun 14, 2021

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