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# arithmetic series

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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

#1
+26222
+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 $$a_1 + a_2 + ... + a_{98} = 99$$.

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

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

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

Jun 14, 2021

#1
+26222
+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 $$a_1 + a_2 + ... + a_{98} = 99$$.

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

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

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

heureka Jun 14, 2021
#2
+121063
+1

Very nice , heureka   !!!!!

CPhill  Jun 14, 2021
#3
+26222
+1

Thank you, CPhill !

heureka  Jun 14, 2021