Question

Find the first term and the difference in an arithmetic sequence if the 100th term is 13 and the 200th term is 82. d = ______(simplify the answer using an integer or simplified fraction)

Answer 1

We will have the following:

*First: We know that arithmetic sequences follow:

$a_n=a_1+d(n-1)$

*Second: From the information given we will have:

a100:

$13=a_1+d(100-1)\Rightarrow13=a_1+99d$

a200:

$82=a_1+d(200-1)\Rightarrow82=a_1+199d$

Then we will find the common difference:

$d=\frac{82-13}{200-100}\Rightarrow d=\frac{69}{100}\Rightarrow d=0.69$

So, the common difference is 0.69.

*Third: We determine the first term:

$13=a_1+0.69(99)\Rightarrow_{}13=a_1+68.31$$\Rightarrow a_1=-55.31\Rightarrow a_1=-\frac{5531}{100}$

So, the first term is -55.31.