Answer:
Yes, if the first term of a geometric sequence is positive and r > 1, then the sequence increases
Step-by-step explanation:
* Lets talk about the geometric sequence
- There is a constant ratio between each two consecutive numbers
- Ex:
# 5 , 10 , 20 , 40 , 80 , ………………………. (×2)
# 5000 , 1000 , 200 , 40 , …………………………(÷5)
* General term (nth term) of a Geometric sequence:
# U1 = a , U2 = ar , U3 = ar2 , U4 = ar3 , U5 = ar4
# Un = ar^n-1, where a is the first term , r is the constant ratio
between each two consecutive terms, and n is the position of
the number in the sequence
- V.I.N: The position of the number means the place of the
number like first , second , third , .......... so n must be positive integer
* Lets talk about the ratio r
- If r greater than 1 and a is positive, the sequence increases lets
take some different examples to explain that
# If the first term is 2 and the ratio between the consecutive
terms is 3/2, then the first four terms in the sequence are
∵ a = 2
∵ r = 3/2 ⇒ greater than 1
∴ First = a = 2
∴ Second = ar = 2 × 3/2 = 3
∴ Third = ar² = 2 × (3/2)² = 2 × 9/4 = 9/2 4.5
∴ Fourth = ar³ = 2 × (3/2)³ = 2 × 27/8 = 27/4 = 6.75
- From the answers the sequence increases
# If the first term is 1/2 and the ratio between the consecutive
terms is 4/3, then the first four terms in the sequence are
∵ a = 1/2
∵ r = 4/3 ⇒ greater than 1
∴ First = a = 1/2
∴ Second = ar = 1/2 × 4/3 = 2/3 ⇒ 2nd > 1st
∴ Third = ar² = 1/2 × (4/3)² = 2 × 16/9 = 8/9 ⇒ 3rd > 2nd
∴ Fourth = ar³ = 1/2 × (4/3)³ = 2 × 64/27 = 32/27 ⇒ 4th > 3rd
- From the answers the sequence increases
* Now we are sure if the first term of a geometric sequence is
positive and r > 1, then the sequence increases
is
is a root, then
is also a root.