Question

In a particular sequence, each term after the first two terms is the sum of the two preceding terms. If the first term is 20 and the sixth term is 220, what is the value of the seventh term? In a particular sequence, each term after the first two terms is the sum of the two preceding terms. If the first term is 20 and the sixth term is 220, what is the value of the seventh term? @Mathematics

Answer 1

To be general, call the first two terms \[a_1,a_2\]. Then we have that
\[a_3 = a_1 + a_2 \]
\[ a_4=a_3+a_2=a_1+2a_2\]
\[ a_5=a_3+a_4=2a_1+3a_2\]
\[a_6=a_4+a_5=3a_1+5a_2\]
\[a_7 = a_5+a_6=5a_1+8a_2\]
since we know that
\[a_1=20\]
and
\[a_6 = 3a_1+5a_2=60+5a_2=220\]
it follows that
\[a_2=\frac{220-60}{5} =32 \]
and so
\[a_7 = 5a_1+8a_2=5(20)+8(32) = 356\]