Question

The 2nd and 5th term of G.P are 2/3 and 1/2 respectively.what is the 1st term,the 4th term and the 7th term

Answer 1

Help me

Answer 2

Xn = a r^(n-1)

X2 = a r = 2/3
X5 = a r^4 = 1/2
so
X5/X2 = r^4/r = (1/2)/(2/3)
r^3 = 3/4
r= .75^(1/3)

a r = 2/3
a [.75^(1/3)] = 2/3
solve for a which is the first term
ar^3 = a * (2/3) which is term 4

Answer 3

1 WHOLE NUMBER 1 OVER 3

Answer 4

[AI-generated answer — Explain Bot]

To find the 1st term, 4th term, and 7th term of a geometric progression (G.P), we can use the formula:

𝑎𝑛 = 𝑎₁ × 𝑟^(𝑛−1)

where 𝑎₁ represents the 1st term, 𝑟 represents the common ratio, and 𝑛 represents the term number.

Given that the 2nd term of the G.P is 2/3 and the 5th term is 1/2, we can create the following equations:

𝑎₂ = 𝑎₁ × 𝑟
𝑎₅ = 𝑎₁ × 𝑟⁴

Substituting the given values:

2/3 = 𝑎₁ × 𝑟 ---(1)
1/2 = 𝑎₁ × 𝑟⁴ ---(2)

Now, we can solve this system of equations to find 𝑎₁ and 𝑟.

Step 1: Solve equation (1) for 𝑎₁ in terms of 𝑟:
From equation (1), 𝑎₁ = (2/3) ÷ 𝑟

Step 2: Substitute the value of 𝑎₁ from Step 1 into equation (2):
1/2 = [(2/3) ÷ 𝑟] × 𝑟⁴

Simplifying equation (2):

1/2 = (2/3) × 𝑟³ ---(3)

Step 3: Solve equation (3) for 𝑟:
Multiply both sides of equation (3) by 3:
(1/2) × 3 = 2 × 𝑟³

3/2 = 2𝑟³

Divide both sides by 2:
3/4 = 𝑟³

Take the cube root of both sides:
∛(3/4) = 𝑟

Simplify:
𝑟 = ∛3/∛4

Step 4: Now that we know the value of 𝑟, we can substitute it back into equation (1) to find 𝑎₁:
From equation (1), 2/3 = 𝑎₁ × (∛3/∛4)

Multiply both sides by ∛4:
(2/3) × ∛4 = 𝑎₁ × (∛3/∛4) × ∛4

Simplifying:
2∛4/3 = 𝑎₁ × (∛3)

Divide both sides by ∛3:
(2∛4/3) ÷ (∛3) = 𝑎₁ × (∛3) ÷ (∛3)

Simplifying:
2∛4/3∛3 = 𝑎₁

Step 5: Simplify the value of 𝑎₁:
Multiplying the radicals:
2∛(4×3) / 3∛(3×3) = 𝑎₁

Simplifying:
2∛12 / 3∛9 = 𝑎₁

Further simplifying:
2∛(2×2×3) / 3∛(3×3) = 𝑎₁

Simplifying:
2(2)∛3 / 3(3)∛1 = 𝑎₁

Finally:
4∛3 / 9 = 𝑎₁

Now, we can substitute the values of 𝑎₁ and 𝑟 into the formula for the G.P to find the 1st, 4th, and 7th terms:

To find the 1st term:
𝑎₁ = 4∛3 / 9

To find the 4th term:
𝑎₄ = (4∛3 / 9) × 𝑟³

To find the 7th term:
𝑎₇ = (4∛3 / 9) × 𝑟⁶

Substituting the value of 𝑟 (∛3/∛4) into the above equations will give you the final values of 𝑎₁, 𝑎₄, and 𝑎₇.