Question

№1 S₅:b(n) b₃=18 q=3 №2 (an) d=4 s₅₀=5500 a1=? №3 (an) a₁=10 S₁₄=1050 d-?

Answer 1

1. b1•q^2=18
b1•9=18
b1=18/9=2

b1•(q^5–1) 2•242
S5 = --------------- = ---------- = 242
q–1 2

2a1+49d
2. S50 =------------•50=(2a1+49•4)•25=
2

= (2a1+196)•25

(2a1+196)•25=5500 |:25
2a1+196=220
2a1=24
a1=12

2a1+13d
3. S14 = ------------ • 14 = (2a1+13d)•7 =
2

= (2•10+13d)•7 = (20+13d)•7

(20+13d)•7 = 1050 |:7
20+13d = 150
13d = 130
d = 10

Answer 2

It looks like you have a series of three related sequences, and you're looking to find certain values within each sequence. Let's break it down:

Sequence #1: S₅:b(n) b₃=18 q=3

• S₅ represents the sum of the first 5 terms of sequence b(n).
• b₃ = 18 indicates that the third term of sequence b(n) is 18.
• q = 3 indicates that the common difference between consecutive terms of sequence b(n) is 3.

To find the sum S₅ of the first 5 terms of sequence b(n), you can use the formula for the sum of an arithmetic series:

S₅ = (n/2)[2a₁ + (n-1)d]

where n is the number of terms, a₁ is the first term, and d is the common difference.

In this case, n = 5, a₁ = ? (unknown), and d = 3. We also know that b₃ = 18, so we can use this information to find a₁:

b₃ = a₁ + 2d 18 = a₁ + 2(3) 18 = a₁ + 6

Now, solve for a₁:

a₁ = 18 - 6 a₁ = 12

Now that we have a₁, we can find S₅:

S₅ = (5/2)[2(12) + (5-1)(3)] S₅ = (5/2)[24 + 4(3)] S₅ = (5/2)[24 + 12] S₅ = (5/2)[36] S₅ = 90

So, the sum of the first 5 terms of sequence b(n) is 90.

Sequence #2: (an) d=4 s₅₀=5500 a₁=?

• In this sequence, the common difference d is given as 4.
• s₅₀ = 5500 indicates that the 50th term of sequence an is 5500.
• You need to find a₁, the first term of the sequence.

You can use the formula for an arithmetic sequence to find a₁:

a₅₀ = a₁ + (50 - 1)d

Substitute the values:

5500 = a₁ + 49 * 4

Now, solve for a₁:

5500 = a₁ + 196

a₁ = 5500 - 196 a₁ = 5304

So, the first term of sequence (an) is 5304.

Sequence #3: (an) a₁=10 S₁₄=1050 d=?

• In this sequence, a₁ is given as 10.
• S₁₄ = 1050 represents the sum of the first 14 terms of sequence an.
• You need to find the common difference d.

You can use the formula for the sum of an arithmetic series to find d:

S₁₄ = (14/2)[2a₁ + (14-1)d]

Substitute the values:

1050 = (14/2)[2(10) + (14-1)d]

Now, solve for d:

1050 = (7)[20 + 13d]

Divide both sides by 7:

150 = 20 + 13d

Subtract 20 from both sides:

130 = 13d

Now, divide by 13 to find d:

d = 130 / 13 d = 10

So, the common difference in sequence (an) is 10.