G EOMETRIC S ERIES Unit 1 Sequences and Series

R ECALL  S ERIES : D EFINITION The expression for the sum of the terms of a sequence. G EOMETRIC S ERIES : D EFINITION An expression for the sum of the terms of a geometric sequence,

RECALL If a sequence of values follows a pattern of multiplying a fixed amount (not zero) by each term to arrive at the following term, it is referred to as a geometric sequence. The number multiplied each time is constant (always the same). The fixed amount multiplied is called the common ratio, r, referring to the fact that the ratio (fraction) of the second term to the first term yields this common multiple. To find the common ratio, divide the second term by the first term.

E XAMPLES Geometric Sequence Common Ratio ( r ) 5, 10, 20, 40, , 22, -44, 88,... r = 2 r = -2 To find the sum of a certain number of terms of a FINITE geometric sequence: where S n is the sum of n terms, a 1 is the first term, and r is the common ratio.

U SING THE GEOMETRIC SERIES FORMULA Evaluate the series for the sequence: 3,6,12,24,48,96 r = 2 o What’s the common ratio r ? o What’s a 1 ? a 1 = 3 o What’s n ? n = 6

E VALUATE :

Y OU TRY ! Identify a 1, r, and n for each sequence. Then evaluate each series.

C ONVERGENCE A geometric series can CONVERGE (getting closer and closer) to the sum, S. OR a geometric series can DIVERGE (does not approach any number… it just keeps going forever)

E XAMPLE A geometric series will converge iff: Example: A geometric series will diverge iff:

I F AN INFINITE SERIES CONVERGES We can find the sum! If Example:  The length of the outside shell is 0.9 times the length of the inside (chambered nautilus). Estimate the total length of the outside shell for the enclosed chambers. The outside edge of the largest side is 27mm.

H OMEWORK Work on project! Bring materials to finish it on Monday Hand in Homework 1 is Due on Tuesday