10:12 C LTE \( \square \) ) docs.google.com Option 4 3. * 20 points The third term in an arithmetic sequence is 58 and the fifth term is 90 . If the first term is \( a_{1} \), which is an equation for the nth term of this sequence? A. \( a_{n}=8 n+10 \) B. \( a_{n}=8 n-14 \) C. \( a_{n}=16 n+10 \) D. \( a_{n}=16 n-38 \) Option 1 Option 2 Option 3 Option 4 4. * 20 points An arithmetic sequence has a first term of 10 and a sixth term of 40 . What is the 20 th term of this sequence? A. 105 B. 110 C. 124 D. 130 Option 1 Option 2 Option 3 Option 4 5. * 20 points ! What is the common anmerence on une anumetre sequence below?

Let's solve the problems step by step. ### Problem 1: Finding the nth term of the arithmetic sequence We know: - The third term \( a_3 = 58 \) - The fifth term \( a_5 = 90 \) In an arithmetic sequence, the nth term can be expressed as: \[ a_n = a_1 + (n-1)d \] where \( a_1 \) is the first term and \( d \) is the common difference. From the information given: 1. For the third term: \[ a_3 = a_1 + 2d = 58 \quad \text{(1)} \] 2. For the fifth term: \[ a_5 = a_1 + 4d = 90 \quad \text{(2)} \] Now, we can solve these two equations simultaneously. Subtract equation (1) from equation (2): \[ (a_1 + 4d) - (a_1 + 2d) = 90 - 58 \] This simplifies to: \[ 2d = 32 \implies d = 16 \] Now substitute \( d = 16 \) back into equation (1): \[ a_1 + 2(16) = 58 \] \[ a_1 + 32 = 58 \implies a_1 = 26 \] Now we can write the nth term: \[ a_n = a_1 + (n-1)d = 26 + (n-1) \cdot 16 \] \[ a_n = 26 + 16n - 16 = 16n + 10 \] Thus, the equation for the nth term is: \[ a_n = 16n + 10 \] The correct option is **C. \( a_n = 16n + 10 \)**. ### Problem 2: Finding the 20th term of the arithmetic sequence We know: - The first term \( a_1 = 10 \) - The sixth term \( a_6 = 40 \) Using the nth term formula: \[ a_n = a_1 + (n-1)d \] For the sixth term: \[ a_6 = a_1 + 5d = 40 \quad \text{(3)} \] Substituting \( a_1 = 10 \) into equation (3): \[ 10 + 5d = 40 \] \[ 5d = 30 \implies d = 6 \] Now we can find the 20th term: \[ a_{20} = a_1 + (20-1)d = 10 + 19 \cdot 6 \] \[ a_{20} = 10 + 114 = 124 \] Thus, the 20th term is **C. 124**. ### Problem 3: Finding the common difference The common difference \( d \) in an arithmetic sequence can be found by taking the difference between any two consecutive terms. If you provide the specific terms of the arithmetic sequence, I can help you calculate the common difference. Please provide the terms of the sequence.