Answer :
Final answer:
The given AP is -4, 2, 8, 14, 20, ... The 12th term is 62. The nth term of the given AP is an = -4 + (n-1)*6.
Explanation:
To determine the arithmetic progression (AP) and its 12th term, we need to use the given sum of the first n terms of the AP, which is 3n²-4n. From this expression, we can identify the common difference and the first term of the AP. The sum of an AP can be derived using the formula Sn = n/2 [2a + (n-1)d], where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Comparing this with the given expression 3n²-4n, we can find that a = -4 and d = 6. Therefore, the AP is -4, 2, 8, 14, 20, ..., and the 12th term is obtained by substituting n = 12 into the nth term formula: a + (12-1)d = -4 + 11*6 = 62.
So, the AP is -4, 2, 8, 14, 20, ... and its 12th term is 62. The nth term of an AP can be found using the formula an = a + (n-1)d, where an is the nth term, a is the first term, and d is the common difference. Therefore, the nth term of the given AP is an = -4 + (n-1)*6.
The sum of the first n terms, Sn, of an Arithmetic Progression (AP) is given by Sn = 3n²- 4n. The formula for the sum of first n terms of an AP is Sn = n/2 [2a + (n - 1) d], where a is the first term, and d is the common difference. By comparing these two formulas, we can find out that 2a = -4 and (n-1)d = 3n, giving a = - 2 and d = 3.
Then the nth term of the AP can be given by the formula An = a + (n - 1) d = -2 + (n-1) * 3 = 3n - 5. Therefore, the 12th term will be substituted with n in the formula, yielding 3(12) - 5 = 31.
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