Question

The following is an arithmetic sequence that has 10 numbers LaTeX: 1,\ a_2, \ a_3, \ a_4, \ \cdots,\ a_{10}.1 , a 2 , a 3 , a 4 , ⋯ , a 10 . If these 10 numbers add up to 100, what is the second number LaTeX: a_2a 2?

Answer 1

Answer:

$a_2=3$

Step-by-step explanation:

The arithmetic sequence is given as: $1,\ a_2, \ a_3, \ a_4, \ \cdots,\ a_{10}$

We are told that the 10 numbers add up to 100.

$S_n=\frac{n}{2}(a_1+l)\\ Where: \\n=number of terms\\l=last term\\a_1 =first term$

$S_{10}=100\\100=\dfrac{10}{2}(1+l)\\100=5(1+l)\\ Divide both sides by 5 \\20=1+l\\ Last term, l=20-1=19$

We know that the nth term of an arithmetic sequence

$l=a_1+(n-1)d\\19=1+(10-1)d\\19=1+9d\\9d=19-1\\9d=18\\d=2$

Therefore, the second number

$a_2=a_1+2\\=1+2\\a_2=3$