What is the 32nd term of the arithmetic sequence where a₁ = 14 and a₁₃ = -58?

Solution:

Given: First term a = a₁ = 14

The common difference is not known.

To find:32^nd term = a₃₂

The n^th term of an arithmetic sequence is given by a_n = a + (n - 1)d

To find the common difference 'd' substitute in the formula, we get

a₁₃ = -58

⇒ a + (13 - 1) d = -58

⇒ 14 + 12d = -58

⇒ 12d = -58 -14

⇒ 12d = -72

⇒ d = -6

Now find a₃₂

a₃₂ = a + (32 -1 ) × d

a₃₂ = 14 + 31 × (-6)

a₃₂ = 14 + (-186)

a₃₂ = -172

Therefore, the value of a₃₂ = -172.

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What is the 32nd term of the arithmetic sequence where a1 = 14 and a13 = -58?

Summary:

The 32nd term of the arithmetic sequence where a₁ = 14 and a₁₃ = -58 is a₃₂ = -172.