Question 16 Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Class X1 - Maths -Sequences and Series Page 192
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Let a, ar, ar², ar³, ...... are in GP
A/C to question,
Sum of first two terms = -4
e.g a + ar = -4 _______(1)
and fifth term ( T₅) = 4 × 3rd term (T₃)
e.g. ar⁵⁻¹ = 4 × ar³⁻¹
⇒ar⁴ = 4ar²
⇒r² = 4
⇒r = ±2
put r = ±2 in equation (1)
a + ar = -4
a + a(±2) = -4
a - 2a = -4 or , a + 2a = -4
-a = -4 or, 3a = -4
a = 4 or, a = -4/3
Hence,
There are two possible series
-4 , 8, -16, 32,..... ( When r = -2)
-4/3 , -8/3, -16/3, ....( When r = 2 )
A/C to question,
Sum of first two terms = -4
e.g a + ar = -4 _______(1)
and fifth term ( T₅) = 4 × 3rd term (T₃)
e.g. ar⁵⁻¹ = 4 × ar³⁻¹
⇒ar⁴ = 4ar²
⇒r² = 4
⇒r = ±2
put r = ±2 in equation (1)
a + ar = -4
a + a(±2) = -4
a - 2a = -4 or , a + 2a = -4
-a = -4 or, 3a = -4
a = 4 or, a = -4/3
Hence,
There are two possible series
-4 , 8, -16, 32,..... ( When r = -2)
-4/3 , -8/3, -16/3, ....( When r = 2 )
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