Question

If the first term of a geometric sequence is positive, and r>1, then the sequnce increases?

Answer 1

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

Answer 2

Answer:

The given statement is TRUE.

Step-by-step explanation:

We are given that if the first term of a geometric sequence is positive, and r>1, then the sequence increases which is true.

If the first term of any geometric sequence is positive and its common ratio ( r ) is greater than 1 then the sequence will always increase.

$a _ n = a r ^ n - 1$