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The 11th term of an arithmetic progression is 14 and the sum of the first 26 terms is 416. Find the first term and the common difference.​

1 Answer

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Answer:

The first term is 6; the common difference in 0.8.

Explanation:

The nth term is:


a_n = a_1 + (n - 1)d

The sum of the first n terms is:


S_n = (n(a_1 + a_n))/(2)


a_(n) = a_1 + (n - 1)d


a_(11) = a_1 + (11-1)d


a_1 + 10d = 14 Equation 1


S_n = (n(a_1 + a_n))/(2)


S_(26) = (26(a_1 + a_(26)))/(2)


(26(a_1 + a_1 + 25d)/(2) = 416


(52a_1 + 650d))/(2) = 416


26a_1 + 325d = 416 Equation 2

Equation 1 and Equation 2 form a system of equations in 2 unknowns.

To eliminate a_1, subtract 26 times Eq. 1 from Eq. 2.


65d = 52


d = (52)/(65)


d = (4)/(5) = 0.8


a_1 + 10d = 14


a_1 + 10 * 0.8 = 14


a_1 + 8 = 14


a_1 = 6

Answer:

The first term is 6; the common difference in 0.8.

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