We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
24
1
avatar

a_1, a_2, a_3, ..., a_{99} is an arithmetic progeression.  If a_2 + a_5 + a_8 + ... + a_{98} = 205, then find a_1 + a_2 + a_3 + ... + a_{99}.  Be sure to show all your work.

 Dec 3, 2019
 #1
avatar+23575 
+1

please help
\(a_1, a_2, a_3, \cdots , a_{99}\) is an arithmetic progeression.  
If \(a_2 + a_5 + a_8 + \cdots+ a_{98} = 205\),
then find \(a_1 + a_2 + a_3 + \cdots + a_{99}\).

 

Formula arithmetic progeression: \(a_n = a_1 + (n-1)d \)

 

\(\begin{array}{|lrcll|} \hline a_2 =& a_1 + (2-1)*d &=& a_1 + d \\ a_5 =& a_1 + (5-1)*d &=& a_1 + 4d \\ a_8 =& a_1 + (8-1)*d &=& a_1 + 7d \\ a_{11} =& a_1 + (11-1)*d &=& a_1 + 10d \\ \cdots \\ a_{98} =& a_1 + (98-1)*d &=& a_1 + 97d \\ \hline \text{terms:} & 2+3(i-1) = 98 \\ & 3(i-1) = 96 \\ & i-1 = 32 \\ & i = 33 \\ \hline \text{sum}& 205&=& 33*a_1+d(1+4+7+10+\cdots + 97 ) \\ & 205&=& 33*a_1+d\left(\dfrac{1+97}{2}\right)\times 33 \\ &\mathbf{ 205 }&=& \mathbf{ 33*a_1+49\cdot 33d } \\ \hline \end{array} \)

 

\(\begin{array}{|lrcll|} \hline a_1 =& a_1 &=& a_1 \\ a_2 =& a_1 + (2-1)*d &=& a_1 + d \\ a_3 =& a_1 + (3-1)*d &=& a_1 + 2d \\ a_4 =& a_1 + (4-1)*d &=& a_1 + 3d \\ a_5 =& a_1 + (5-1)*d &=& a_1 + 4d \\ \cdots \\ a_{99} =& a_1 + (99-1)*d &=& a_1 + 98d \\ \hline \text{terms:} & 99 \\ \hline & \text{sum} &=& 99*a_1+d(1+2+3+4+\cdots + 98 ) \\ & \text{sum}&=& 99*a_1+d\left(\dfrac{1+98}{2}\right)\times 98 \\ &\mathbf{ \text{sum} }&=& \mathbf{ 99*a_1+49\cdot 99d } \\ & \text{sum} &=& 3\times \left( \underbrace{33*a_1+49\cdot 33d}_{=205} \right) \\ & \text{sum} &=& 3\times 205 \\ & \mathbf{ \text{sum} } &=& \mathbf{615} \\ \hline \end{array}\)

 

\(a_1 + a_2 + a_3 + \cdots + a_{99} = 615 \)

 

laugh

 Dec 3, 2019

28 Online Users

avatar