Answer:

The 15ᵗʰ term of AP is 80.

Step-by-step-explanation:

We have GIVEN that,

The sum of first 20 TERMS of an A.P. is 1060.

The 5ᵗʰ term of AP is 20.

Now, we know that,

Sₙ = n / 2 [ 2a + ( n - 1 ) * d ] - - [ Formula ]

⇒ S₂₀ = 20 / 2 [ 2a + ( 20 - 1 ) * d ]

⇒ 1060 = 10 ( 2a + 19d )

⇒ 2a + 19d = 1060 ÷ 10

⇒ 2a + 19d = 106 - - ( 1 )

Now, we know that,

tₙ = a + ( n - 1 ) * d - - [ Formula ]

⇒ t₅ = a + ( 5 - 1 ) * d

⇒ 20 = a + 4d

⇒ a + 4d = 20 - - ( 2 )

By multiplying equation ( 2 ) by 2, we get,

⇒ 2 × ( a + 4d ) = 20 × 2

⇒ 2a + 8d = 40 - - ( 3 )

By subtracting equation ( 3 ) from equation ( 1 ), we get,

⇒ 2a + 19d - 2a - 8d = 106 - 40

⇒ 11d = 66

⇒ d = 66 ÷ 11

⇒ d = 6

By SUBSTITUTING d = 6 in equation ( 2 ), we get,

⇒ a + 4d = 20 - - ( 2 )

⇒ a + 4 ( 6 ) = 20

⇒ a + 24 = 20

⇒ a = 20 - 24

⇒ a = - 4

Now, by using the formula tₙ = a + ( n - 1 ) * d,

⇒t₁₅ = a + ( 15 - 1 ) * d

⇒ t₁₅ = ( - 4 ) + 14 × 6

⇒ t₁₅ = - 4 + 84

⇒ t₁₅ = 80

∴ The 15ᵗʰ term of AP is 80.

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Additional Information:

Arithmetic Progression:

1. In a sequence, if the common difference between two CONSECUTIVE terms is constant, then the sequence is called as Arithmetic Progression ( AP ).

2. nᵗʰ term of an AP:

The number of a term in the given AP is called as ] term of an AP.

3. Formula for nᵗʰ term of an AP:

• tₙ = a + ( n - 1 ) * d

4. The sum of the first n terms of an AP:

The ADDITION of either all the terms of a particular terms is called as sum of first n terms of AP.

5. Formula for sum of the first n terms of A. P. :

• Sₙ = n / 2 [ 2a + ( n - 1 ) * d ]