Given a_n = 5+3(n-1).

Welcome back Debby — missed you this week. Step 1: Identify the common difference. The n^th term of an arithmetic progression is given by a_n = a + (n-1)d, where a is the first term and d is the common difference. Given a_n = 5+3(n-1). Comparing this to the general formula, we can see that the common difference d is the coefficient of (n-1). d = 3 Step 2: Find the first term (a_1) and the sixth term (a_6). Substitute n=1 into the given formula to find the first term: a_1 = 5+3(1-1) = 5+3(0) = 5 Substitute n=6 into the given formula to find the sixth term: a_6 = 5+3(6-1) = 5+3(5) = 5+15 = 20 Step 3: Calculate the sum of the first six terms (S_6). The formula for the sum of the first n terms of an A.P. is S_n = (n)/(2)(a_1 + a_n). For n=6: S_6 = (6)/(2)(a_1 + a_6) S_6 = 3(5 + 20) S_6 = 3(25) S_6 = 75 The common difference is 3 and the sum of the first six terms is 75. The common difference is 3 The sum of the first six terms is 75 Step 1: Rationalize the first term. Multiply the numerator and denominator by the conjugate of the denominator, 5-sqrt(3). sqrt(3)5+sqrt(3) = sqrt(3)5+sqrt(3) × 5-sqrt(3)5-sqrt(3) = sqrt(3)(5-sqrt(3))(5)^2 - (sqrt(3))^2 = 5sqrt(3) - (sqrt(3))^225 - 3 = 5sqrt(3) - 322 Step 2: Rationalize the second term. Multiply the numerator and denominator by the conjugate of the denominator, sqrt(5)+sqrt(3). 2sqrt(5)sqrt(5)-sqrt(3) = 2sqrt(5)sqrt(5)-sqrt(3) × sqrt(5)+sqrt(3)sqrt(5)+sqrt(3) = 2sqrt(5)(sqrt(5)+sqrt(3))(sqrt(5))^2 - (sqrt(3))^2 = 2(sqrt(5))^2 + 2sqrt(5)sqrt(3)5 - 3 = 2(5) + 2sqrt(15)2 = 10 + 2sqrt(15)2 = 5 + sqrt(15) Step 3: Add the rationalized terms. 5sqrt(3) - 322 + (5 + sqrt(15)) To combine, find a common denominator: = 5sqrt(3) - 322 + 22(5 + sqrt(15))22 = 5sqrt(3) - 3 + 110 + 22sqrt(15)22 = 107 + 5sqrt(3) + 22sqrt(15)22 This expression is not in the form a+bsqrt(c) with a single sqrt(c) term. Let's recheck the problem statement. It asks for a+bsqrt(c). This implies c should be a single integer. The current result has sqrt(3) and sqrt(15). This suggests there might be a simplification or an error in my calculation or interpretation. Let's re-examine the student's work on the right side of the image. The student has: -2(5) + 2sqrt(15) = 10 + 2sqrt(15) (This seems to be from the second term's numerator, but the sign is wrong for the first part) sqrt(15-3) + 5sqrt(15) = sqrt(13) - 3 + 10 + 2sqrt(15) (This looks like an attempt to combine terms, but it's very messy) -7+3sqrt(15) 2 Let's re-evaluate my calculation for the second term: 2sqrt(5)sqrt(5)-sqrt(3) = 2sqrt(5)(sqrt(5)+sqrt(3))(sqrt(5)-sqrt(3))(sqrt(5)+sqrt(3)) = 2(5) + 2sqrt(15)5-3 = 10+2sqrt(15)2 = 5+sqrt(15). This is correct. Let's re-evaluate my calculation for the first term: sqrt(3)5+sqrt(3) = sqrt(3)(5-sqrt(3))(5+sqrt(3))(5-sqrt(3)) = 5sqrt(3)-325-3 = 5sqrt(3)-322. This is also correct. So, the sum is 5sqrt(3)-322 + 5+sqrt(15). This is (-3)/(22) + 5sqrt(3)22 + 5 + sqrt(15). This cannot be simplified into the form a+bsqrt(c) where c is a single integer, unless sqrt(3) and sqrt(15) can be combined, which they cannot. Perhaps the question implies that c is a specific value, or that one of the radical terms should disappear. Let's check if there's a typo in the question. If the first term was sqrt(3)5-sqrt(3) or the second term was 2sqrt(3)sqrt(5)-sqrt(3), it might simplify differently. Given the problem as written: sqrt(3)5+sqrt(3) + 2sqrt(5)sqrt(5)-sqrt(3) = a+bsqrt(c). Let's assume c is meant to be 15 or 3. If c=15, then we need to combine terms with sqrt(15). If c=3, then we need to combine terms with sqrt(3). Let's look at the student's work again. The student has a term -7+3sqrt(15)2. This suggests they might have gotten a common denominator of 2 for some part. Let's try to get a common denominator of 22 for the entire expression. 5sqrt(3)-322 + 22(5+sqrt(15))22 = 5sqrt(3)-3+110+22sqrt(15)22 = 107+5sqrt(3)+22sqrt(15)22 = (107)/(22) + (5)/(22)sqrt(3) + (22)/(22)sqrt(15) = (107)/(22) + (5)/(22)sqrt(3) + sqrt(15) This is of the form a+b_1sqrt(c_1)+b_2sqrt(c_2), not a+bsqrt(c). There might be an error in the problem statement or a misunderstanding of the expected form. However, I must solve the question as given. If the question implies that c is a specific value, it's not stated. If c can be either 3 or 15, then the expression is not uniquely a+bsqrt(c). Let's consider the possibility that the question expects c to be the same for both terms after some manipulation, or that one of the radical terms cancels out. This is not happening here. Let's re-check the arithmetic carefully. Term 1: sqrt(3)5+sqrt(3) = sqrt(3)(5-sqrt(3))25-3 = 5sqrt(3)-322 Term 2: 2sqrt(5)sqrt(5)-sqrt(3) = 2sqrt(5)(sqrt(5)+sqrt(3))5-3 = 10+2sqrt(15)2 = 5+sqrt(15) Sum: 5sqrt(3)-322 + 5+sqrt(15) = (-3)/(22) + 5sqrt(3)22 + 5 + sqrt(15) = (5 - (3)/(22)) + 5sqrt(3)22 + sqrt(15) = ((110-3)/(22)) + 5sqrt(3)22 + sqrt(15) = (107)/(22) + (5)/(22)sqrt(3) + sqrt(15) This result has two different square root terms, sqrt(3) and sqrt(15). If the question implies that c is a single integer, then the problem as stated cannot be simplified to a+bsqrt(c). However, if c is not specified, then the expression is in the form a+b_1sqrt(c_1)+b_2sqrt(c_2). Given the format a+bsqrt(c), it usually implies a single radical term. Let's assume there might be a typo in the question and one of the sqrt(3) or sqrt(5) should be different to allow for simplification. For example, if the first term was sqrt(5)5+sqrt(3), it would still lead to mixed radicals. If the second term was 2sqrt(3)sqrt(5)-sqrt(3), then: 2sqrt(3)(sqrt(5)+sqrt(3))5-3 = 2sqrt(15)+2(3)2 = sqrt(15)+3. Then the sum would be 5sqrt(3)-322 + sqrt(15)+3 = 5sqrt(3)-3+22sqrt(15)+6622 = 63+5sqrt(3)+22sqrt(15)22. Still two radicals. What if c is not an integer? The problem states a, b, c are constants. Usually, c is an integer in these types of problems. Let's consider the possibility that the student's work on the right side is a hint. The student has -7+3sqrt(15)2. If we assume the final form is a+bsqrt(15), then the sqrt(3) term must cancel or be zero. This is not the case. Given the exact wording "find the value of a and b", and the form a+bsqrt(c), it is highly likely that c is a specific integer, and the expression should simplify to have only one radical. Since it doesn't, I will state the result as is, and mention the issue. However, the instructions say "Solve questions completely and accurately." and "Formula → substitute → simplify step by step. Show every step." My steps are accurate for the given expression. Let's re-read the question carefully: "Given that sqrt(3)5+sqrt(3) + 2sqrt(5)sqrt(5)-sqrt(3) = a+bsqrt(c), where a, b and c are constants, find the value of a and b." It does not ask for c. It asks for a and b. This implies c is implicitly determined by the expression. If the expression results in (107)/(22) + (5)/(22)sqrt(3) + sqrt(15), then it cannot be written as a+bsqrt(c) for a single c. Could it be that c is a product of primes, e.g., c=3 × 5 = 15? If c=15, then we need to express sqrt(3) in terms of sqrt(15). sqrt(3) = sqrt(15)sqrt(5). This doesn't help. Let's assume the