Question

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.

Answer 1

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

Answer 2

a1 = 56

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

a1 = 56 & a2 = 49

Answer 3

\[a_n = a_1+(n-1)d\]\[S_n = \frac{n(a_1+a_n)}{2}\] \[a_6 =a_1+(6-1)d =21 \implies \color{red}{21 = a_1+5d}\]\[S_{17} =\frac{17(a_1+\color{blue}{a_{17}})}{2} = 0\ \\ \qquad a_{17}=a_1 +(17-1)d \implies \color{blue}{a_{17} = a_1 +16d}\]\[S_{17} =\frac{17(a_1+a_1+16d)}{2}=0 \\ S_{17} =\frac{17(2a_1+16d)}{2}=0 \\ \qquad \implies \frac{17 \cdot 2(a_1+8d)}{2}=0 \\ \qquad \implies 17(a_1+8d)=0 \\ \qquad \implies a_1+8d = 0 \\ \qquad \implies \color{orange}{a_1 =-8d}\]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.\[\color{pink}{d=-\frac{a_1}{8} = -\frac{56}{8} = -7}\]\[a_2 =\color{purple}{ a_1}+(n-1)\color{pink}{d} \\ a_2=56+(2-1)(-7) \\ a_2=56-7 \\ \color{gold}{a_2=49}\]

Answer 4

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

Answer 5

looks like u got a color wand :D

Answer 6

I had to.... I confused myself otherwise

Answer 7

:)

Answer 8

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