# Thread: Finding 'a' given two summations

1. ## Finding 'a' given two summations

I've been stuck on how to do this question for over an hour. I clearly don't understand how to do it, hopefully somebody can help explain this to me?

Given:

$\sum\limits_{k = 1}^{100}{(6a_k + 8)} = 2000$

$\sum\limits_{j = 0}^{98}{(a_j_+_1)} = 545$

find a100

Thank you.

2. ## Re: Finding 'a' given two summations

Originally Posted by Kevmck
I've been stuck on how to do this question for over an hour. I clearly don't understand how to do it, hopefully somebody can help explain this to me?

Given:

$\sum\limits_{k = 1}^{100}{(6ak + 8)} = 2000$

$\sum\limits_{j = 0}^{98}{(aj+1​)} = 545$

find a100

Thank you.

The formatting makes it impossible to know what you mean.

3. ## Re: Finding 'a' given two summations

Given:

$\sum\limits_{k = 1}^{100}{(6a_k + 8)} = 2000$

$\sum\limits_{j = 0}^{98}{(a_j_+_1 + 5)} = 545$

find a100​

Sorry, I wasn't sure how to fix the formatting before, I shouldn't have left it like that until I figured out how. Fixed.

4. ## Re: Finding 'a' given two summations

Originally Posted by Kevmck
Given:

$\sum\limits_{k = 1}^{100}{(6a_k + 8)} = 2000$

$\sum\limits_{j = 0}^{98}{(a_j_+_1 + 5)} = 545$

find $a_{100}​$

$\sum\limits_{k = 1}^{100} {\left( {6a_k + 8} \right)} = 6\sum\limits_{k = 1}^{100} {a_k } + 8(100)$

$\sum\limits_{j = 0}^{98} {\left( {a_{j + 1} + 5} \right)} = \sum\limits_{j = 1}^{99} {a_j } + 5(99)$

5. ## Re: Finding 'a' given two summations

I'm not sure I really understand, but thank you for the help.

6. ## Re: Finding 'a' given two summations

Hello, Kevmck!

$\text{Given: }\:(1)\;\sum\limits_{k = 1}^{100}{(6a_k + 8)} \:=\: 2000 \qquad (2)\;\sum\limits_{j = 0}^{98}{(a_j_+_1 + 5)} = 545$

$\text{Find }a_{100}.$

$(1)\;\sum^{100}_{k=1}(6a_k + 8) \;=\;2000$

. $6\sum^{100}_{k=1} a_k + \sum^{100}_{k=1}8 \;=\;2000$

. . $6\sum^{100}_{k=1}a_k + 800 \;=\;2000$

. . . . . . $6\sum^{100}_{k=1}a_k \;=\;1200$

. . . . . . . $\sum^{100}_{k=1}a_k \;=\;200\;\;[3]$

$(2) \;\sum^{98}_{j=0}\left(a_{j+1} + 5\right) \;=\;545$

. $\sum^{98}_{j=0}a_{j+1} + \sum^{98}_{j=0}5 \;=\;545$

. . $\sum^{98}_{j=0}a_{j+1} + 495 \;=\;545$

. . . . . . $\sum^{98}_{j=0}a_{j+1} \;=\;50\;\;[4]$

From [3], we have: . $a_1 + a_2 + a_3 + \cdots + a_{99} + a_{100} \:=\:200$

From [4], we have: . $a_1 + a_2 + a_3 + \cdots + a_{99} \qquad\quad =\;\;50$

Subtract: . $a_{100} \:=\:150$

7. ## Re: Finding 'a' given two summations

Thank you Soroban. In writing out potential solutions after getting the help from Plato, I eventually came up with 200 and 50 but I didn't know what I was supposed to do with them from there. This example has helped solidify what was happening and what the question was asking for. Thank you again.