## chillhill one year ago Find the first two terms of an arithmetic sequence if the sixth term is 21 and the sum of the first 17th terms is 0.

1. mathstudent55

Arithmetic sequence: a1 = a1 a2 = a1 + d a3 = a1 + 2d a4 = a1 + 3d a5 = a1 + 4d a6 = a1 + 5d The 6th term, a6, equal a1 + 5d. We know the 6th term is 21, so we get this equation: 21 = a1 + 5d a1 + 5d = 21 (Here is Eq. 1) Sum: Sn = [n(a1 + an)]/2 S17 = [17(a1 + a17)]/2 a17 = a1 + 16d S17 = [17(a1 + a1 + 16d)]/2 We are told that the sum of the first 17 term is 0, so we get the second equation. 0 = [17(a1 + a1 + 16d)]/2 0 = 17(2a1 + 16d)/2 0 = 17a1 + 136d 17a1 + 136d = 0 (Here is the 2nd equation) Now we solve these equations simultaneously: a1 + 5d = 21 (Eq. 1) 17a1 + 136d = 0 (Eq. 2) Multiply Eq. 1 by -17 and add to Eq. 2: 51d = -357 d = -7 a1 + 5(-7) = 21 a1 - 35 = 21 a1 = 56

2. mathstudent55

a1 = 56 a2 = a1 + d a2 = 56 + (-7) = 49 a1 = 56 & a2 = 49

3. anonymous

$a_n = a_1+(n-1)d$$S_n = \frac{n(a_1+a_n)}{2}$ $a_6 =a_1+(6-1)d =21 \implies \color{red}{21 = a_1+5d}$$S_{17} =\frac{17(a_1+\color{blue}{a_{17}})}{2} = 0\ \\ \qquad a_{17}=a_1 +(17-1)d \implies \color{blue}{a_{17} = a_1 +16d}$$S_{17} =\frac{17(a_1+a_1+16d)}{2}=0 \\ S_{17} =\frac{17(2a_1+16d)}{2}=0 \\ \qquad \implies \frac{17 \cdot 2(a_1+8d)}{2}=0 \\ \qquad \implies 17(a_1+8d)=0 \\ \qquad \implies a_1+8d = 0 \\ \qquad \implies \color{orange}{a_1 =-8d}$We can use this to find d, then plug it into the red equation. $\color{orange}{a_1=-8d} \implies \color{green}{d=-\frac{a_1}{8}}$$21=a_1+5\left(\color{green}{-\frac{a_1}{8}}\right)$$21=a_1-\frac{5a_1}{8}\qquad \implies 21=a_1\left(1-\frac{5}{8}\right) \qquad \implies 21 = \frac{3}{8}a_1$$\color{purple}{a_1=56}$Now that we've found our first term, we can use that to find the difference, d, and then our second term using the formula for arithmetic sequence.$\color{pink}{d=-\frac{a_1}{8} = -\frac{56}{8} = -7}$$a_2 =\color{purple}{ a_1}+(n-1)\color{pink}{d} \\ a_2=56+(2-1)(-7) \\ a_2=56-7 \\ \color{gold}{a_2=49}$

4. mathstudent55

@Jhannybean That is beautiful. I'm almost in tears just looking at it!!! Just look at the use of colors!

5. imqwerty

looks like u got a color wand :D

6. anonymous

I had to.... I confused myself otherwise

7. imqwerty

:)

8. imqwerty

i make circles on useful results nd info to note them ( ͡° ͜ʖ ͡°)