Grade 12 Number Patterns: The first, third and thirteenth terms of an arithmetic sequence are the first 3 terms of a geometric sequence. If the first term of both sequences is 1, determine:

1.) the first three terms of the geometric sequence if r > 1

2.) the sum of 7 terms of the geometric sequence if the sequence is 1, 5, 25​

1. The first three terms of the geometric sequence are 1, 5, 25

2. Sum of the 7th terms of the geometric sequence is 19531

Step-by-step explanation:

We'll begin by calculating the common ratio (r). This can be obtained as follow:

For Arthmetic sequence:

First term (a) = 1

3rd term (T₃) = a + 2d

3rd term (T₃) = 1 + 2d

13th term (T₁₃) = a + 12d

13th term (T₁₃) = 1 + 12d

For Geometric sequence:

First term (a) = 1

2nd term (T₂) = ar

2nd term (T₂) = 1 × r

2nd term (T₂) = r

3rd term (T₃) = ar²

3rd term (T₃) = 1 × r²

3rd term (T₃) = r²

From the question given above,

The first term of the arithmetic sequence is also the first term of the geometric sequence.

The 3rd term of the arithmetic sequence is the 2nd term of the geometric sequence i.e

3rd term (T₃) = 2nd term (T₂)

1 + 2d = r

The 13th term of arithmetic sequence is the 3rd term of the geometric sequence i.e

13th term (T₁₃) = 3rd term (T₃)

1 + 12d = r²

SUMMARY:

1 + 2d = r (1)

1 + 12d = r² (2)

Substitute value of r from equation 1 into equation 2

1 + 12d = (1 + 2d)²

1 + 12d = 1 + 2d + 2d + 4d²

1 + 12d = 1 + 4d + 4d²

Rearrange

1 – 1 + 4d – 12d + 4d² = 0

–8d + 4d² = 0

4d² – 8d = 0

4d(d – 2) = 0

d = 0 or 2

Since the common difference (d) can not be 0, thus, is 2

Substitute the value of d into equation 1

1 + 2d = r

d = 2

1 + 2(2) = r

1 + 4 = r

r = 5

Thus, the common ratio (r) is 5

1. Determination of the first three terms of the geometric sequence.

First term (a) = 1

Common ratio (r) = 5

2nd term (T₂) = ar

2nd term (T₂) = 1 × 5

2nd term (T₂) = 5

3rd term (T₃) = ar²

3rd term (T₃) = 1 × 5²

3rd term (T₃) = 1 × 25

3rd term (T₃) = 25

Thus, the first three terms of the geometric sequence are 1, 5, 25

2. Determination of the sum of 7th terms of the geometric sequence .

1, 5, 25

First term (a) = 1

Common ratio (r) = 5

Number of term (n) = 7

Sum of the 7th term (S₇) =?

Sₙ = a[rⁿ – 1]/r – 1

S₇ = 1[5⁷ – 1]/5 – 1

S₇ = [78125 – 1]/4

S₇ = 78124 / 4

S₇ = 19531

Thus, sum of the 7th terms of the geometric sequence is 19531

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