The 3rd term of a geometric sequence is -2 and the 7th is -32. find the common ratio, the first term, the explicit formula, and the 10th term.

see explanation

Step-by-step explanation:

The n th term of a geometric sequence is

= a

where a is the first term and r the common ratio

Both a and r have to be found

Given a₃ = - 2, then

ar² = - 2 → (1)

Given a₇ = - 32, then

a = - 32 → (2)

Divide (2) by (1)

= , that is

= 16 ( take the fourth root of both sides )

r = 2 ← common ratio

Substitute r = 2 into (1)

a × 2² = - 2, that is

4a = - 2 ( divide both sides by 4 )

a = - ← first term

Hence

= - ← explicit formula

and

= - × = - 0.5 × 512 = - 256

given that triangle abc is transformed two times to get triangle

a"b"c"

first abc is transformed in to a'b'c' as follows:

the coordinates of a are (1,-1) and transformed into a'(-1,1)

similarly b (4,-2) became b'(-4,2) and

c(7,-2) became c'(-7,2)

i.e. (x,y) becomes (-x,-y)

this is nothing but reflection about a point here origin.

thus first transformation is reflection on the origin.

next is exactly shifting 3 units down

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