# Thread: sum of an arithmetic progression

1. ## sum of an arithmetic progression

The ratio of the 5th term to the 12th term of a sequence in an arithmetic progression is 6/13. If each of this sequence is positive, and the product of the 1rst term and the 3rd term is 32, find the sum of the first 100 terms of this sequence.

Hope someone can help, I'm studying for my math final. Which formula should I use?

2. You are told that $\displaystyle \dfrac{U_5}{U_{12}} = \dfrac{6}{13}$ and $\displaystyle U_1U_3 = 32$.

By definition $\displaystyle a = U_1$ and $\displaystyle U_n = a+(n-1)d$

Next work out $\displaystyle U_5$ and $\displaystyle U_{12}$ and $\displaystyle U_3$ in terms of $\displaystyle a$ and $\displaystyle d$ and the solve the simultaneous equations

Edit: Sum of an arithmetic sequence is $\displaystyle S_n = \dfrac{n}{2}\left(2a+(n-1)d\right)$

3. Before finding $\displaystyle \displaystyle S_{100}$ you need to find $\displaystyle \displaystyle a$ and $\displaystyle \displaystyle d$ such that

$\displaystyle \displaystyle\frac{t_5}{t_{13}}=\frac{6}{13}=\frac {a+(5-1)d}{a+(13-1)d}$

and

$\displaystyle \displaystyle t_1\times t_3 = t_3 \times a = 32$

Does this help?

4. Hey, thank you so much for helping me but I'm still not getting it, could you please explain further?

5. Originally Posted by stressedIBstudent1
Hey, thank you so much for helping me but I'm still not getting it, could you please explain further?
You should know that $\displaystyle U_5 = a+4d$ and $\displaystyle U_{12} = a + 11d$

$\displaystyle \dfrac{a+4d}{a+11d} = \dfrac{6}{13} \text{.......... eq(1) }$

$\displaystyle U_1U_3 = a(a+2d) = 32 \text{.......... eq(2) }$

Solve simultaenously for a and d.

6. Originally Posted by stressedIBstudent1
The ratio of the 5th term to the 12th term of a sequence in an arithmetic progression is 6/13. If each of this sequence is positive, and the product of the 1rst term and the 3rd term is 32, find the sum of the first 100 terms of this sequence.

Hope someone can help, I'm studying for my math final. Which formula should I use?
$\displaystyle T_1=a,\;\;T_2=a+d,\;\;T_3=a+2d,\;\;T_4=a+3d,\;\;T_ 5=a+4d,\;\;.....T_{12}=a+11d$

$\displaystyle \displaystyle\frac{T_5}{T_{12}}=\frac{a+4d}{a+11d} =\frac{6}{13}\Rightarrow\ 13(a+4d)=6(a+11d)$

$\displaystyle \Rightarrow\ 13a+52d=6a+66d\Rightarrow\ 13a-6a=66d-52d\Rightarrow\ 7a=14d\Rightarrow\ a=2d$

Substitute this "a" into $\displaystyle a(a+2d)=32$

Alternatively work the other way...

$\displaystyle 32=1(32)=2(16)=4(8)$ so you could test $\displaystyle a=2,\;\; d=7$ and $\displaystyle a=4,\;\;d=2$ in the ratio equation.