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?

Answer 1

Hi,

356

Explanation:

`Let's\ denote\ the\ sequence\ as\ a_1,a_2,a_3,a_4,a_5,a_6,a_7 ,...\\where\ a_1=20\ and\ a_6=220.\\\\\text{We know that each term after the first two terms is the sum of the two preceding terms.}\\\text{Using this information, we can find the next terms until we reach the seventh term.}\\We\ have:a_1=p\\a_2=q\\a_3=a_1+a_2=p+q\\a_4=a_2+a_3=q+p+q=p+2q\\a_5=a_3+a_4=p+q+p+2q=2p+3q\\a_6=a_4+a_5=p+2q+2p+3q=3p+5q\\`

`Given\ that\ p=20\ and\ a_6=220=3*20+5q\\we\ can\ solve\ for\ q :\\5q+60=220\\5q=160\\q=32=a_2\\Now\ that\ we\ have :\\a_7=a_5+a_6=2p+3q+3p+5q=5p+8q=5*20+8*32=356\\\text{So, the seventh term in the sequence is 356.}\\`