The sum of six terms of a geometric sequence is -63. If the first term in 3, find the commob ratio. Pa help po pls asap
Answers
Answer:
r = -2
Step-by-step explanation:
Given
Sum of the six terms of a G.P = -63
a = 3
Let the common ratio be 'r'.
We know that
Sum of 'n' terms in a G.P with common ratio 'r' = a(rⁿ - 1)/(r - 1)
Here a = 3, sum = -63, n = 6, r = r
⇒a(rⁿ - 1)/(r - 1) = S
⇒3(r⁶ - 1)/(r - 1) = -63
⇒(r⁶ - 1)/(r - 1) = -63/3
⇒(r⁶ - 1)/(r - 1) = -21
⇒((r²)³ - 1³)/(r - 1) = -21
⇒(r² - 1)(r⁴ + r² + 1)/ (r - 1) = -21 [∵a³- b³ = (a-b)(a² + ab +b²)]
⇒(r - 1)(r + 1)(r⁴ + r²+ 1)/(r - 1) = -21 [∵a² - b² = (a - b)(a + b)]
⇒(r + 1)(r⁴ + r²+ 1) = -21
This equation is only satisfied by r = -2
or
⇒(r + 1)(r⁴ + r²+ 1) = -21
= -1*21
'r⁴ + r²+ 1' gives the greatest and positive value because of the highest degree in it. ( Trail and error method)
So we can equate
r + 1 = -1 _______(1) and r⁴ + r²+ 1 = 21______(2)
From (1)
r + 1 = -1
∴ r = -2
These type of questions are solved by trail and error method.