Find the first term and the common difference of the arithmetic sequence whose 7th term is 19 and 20th term is 71.

Solve the following system:
{19 = 6 A + F | (equation 1)
71 = 19 A + F | (equation 2)

 

Express the system in standard form:
{-F - 6 A = -19 | (equation 1)
-F - 19 A = -71 | (equation 2)

 

Subtract equation 1 from equation 2:
{-F - 6 A = -19 | (equation 1)
0 F - 13 A = -52 | (equation 2)

 

Multiply equation 1 by -1:
{F + 6 A = 19 | (equation 1)
0 F - 13 A = -52 | (equation 2)

 

Divide equation 2 by -13:
{F + 6 A = 19 | (equation 1)
0 F+A = 4 | (equation 2)

Subtract 6 × (equation 2) from equation 1:
{F+0 A = -5 | (equation 1)
0 F+A = 4 | (equation 2)

 

 F = -5 - First term         and A =4 - Common difference