The 10th term in the sequence is 2560 what is the 11th term in the sequence

Full Question:

The 4th term of a g.p. is 40 and the 10th term in the sequence is 2560, what is the 11th term in the sequence ?

the 11 the term is 5120

Step-by-step explanation:

Given

Geometry Progression

4th term = 40

10th term = 2560

Required

11 term.

The nth term of a geometric sequence is calculated as follows

Tₙ = arⁿ⁻¹

For the 4th term, n = 4 and Tₙ = 40

Substitute these in the given formula; this gives

40 = ar⁴⁻¹

40 = ar³. --;; equation 1

For the 10th term, n = 10 and Tₙ = 2560

Substitute these in the given formula; this gives

2560 = ar¹⁰⁻¹

2560 = ar⁹. --;; equation 2

Divide equation 2 by 1. This gives

2560/40 = ar⁹/ar³

64 = r⁹/r³

From laws of indices

64 = r⁹⁻³

64 = r⁶

Find 6th root of both sides

(64)^1/6 = r

r = (2⁶)^1/6

r = 2

Substitute r = 2 in equation 1

40 = ar³. Becomes

40 = a * 2³

40 = a * 8

40 = 8a

Divide both sides by 8

40/8 = 8a/8

5 = a

a = 5.

Now, the 11 term can be solved using Tₙ = arⁿ⁻¹ where n = 11

So,

Tₙ = arⁿ⁻¹ becomes

Tₙ = 5 * 2¹¹⁻¹

Tₙ = 5 * 2¹¹⁻¹

Tₙ = 5 * 2¹⁰

Tₙ = 5 * 1024

Tₙ = 5120.

Henxe, the 11 the term is 5120