Question

In a gp of real number it is given that t1 t2 t3 t4= 30 t1 squares t2 square t3 square t4 sqare=340 determine the first term and common ratios

Answer 1

Answer:

first term  = 2

Common Ratio = 2

Step-by-step explanation:

Let sat first term = a

common ratio = r

t₁ = a   t₁² = a²

t₂ = ar   t₂² = (ar)²

t₃ = ar²   t₃² = (ar²)²

t₄ = ar³   t₄² = (ar³)²

t₁ + t₂ +t₃ + t₄  = a + ar + ar² +  ar³ = 30  => a(1 + r + r² + r³) = 30

=> a(1 + r²)(1 + r) = 30

squaring both sides

=> a²(1 + r²)²(1 + r)² = 900

t₁²+ t₂² + t₃² + t₄²  = a² + (ar)² + (ar²)² + (ar³)² = 340 => a²(1 + r² + r⁴ + r⁶) = 340

=> a²  (1 + r⁴)(1 + r²) = 340

a²(1 + r²)²(1 + r)² / a²  (1 + r⁴)(1 + r²)  = 900/340

=> (1 + r²)(1 + r)² /  (1 + r⁴)  = 45/17

=> 17  (r²+ 1)(r² + 2r + 1)= 45 (1 + r⁴)

=> 17( r⁴ + 2r³ + 2r² + 2r + 1) = 45 + 45r⁴

=> 28r⁴ - 34r³ - 34r² - 34r + 28 =  0

=> (r - 2)(28r³ + 22r²  + 10r  - 14) = 0

=> r = 2

a(1 + r²)(1 + r) = 30

=> a(1 + 4)(1 + 2) = 30

=> a = 2

first term  = 2

Common Ratio = 2

2  + 4 +   8 +  16  = 30

4  + 16 + 64 + 256 = 340