Question

The 12th term of an arithmetic progression has the value 0. Which term of the AP is 3 times the 17th element?​

Answer 1

Answer:

n = 27

Step-by-step explanation:

Given:

• The 12th term of the A.P is 0

To Find:

• Which term of the A.P is 3 times the 17th term

Solution:

The nth term of an A.P is given by,

aₙ = a₁ + (n - 1) × d

where aₙ is the nth term

a₁ is the first term

d is the common difference

By given,

a₁₂ = 0

a₁ + 11d = 0

a₁ = -11d-----(1)

Now let the term which is 3 times the 17th term be aₙ.

Therefore,

aₙ = 3 × (a₁₇)

aₙ = 3 × (a₁ + 16d)

a₁ + (n - 1) × d = 3 × (a₁ + 16d)

Substitute the value of a₁ from equation 1,

-11d + (n - 1) × d = 3 (-11d + 16 d)

-11d + (n - 1) × d = 3 × 5d

-11d + (n - 1) × d = 15 d

(n - 1) × d = 26d

n - 1 = 26d/d

n - 1 = 26

n = 26 + 1

n = 27

Hence the 27th term of the A.P is 3 times the 17th term.

Answer 2

$\bold{to \: find} \\ \longmapsto \tt which \: \: term \: \\ \: of \: \: the \: \: a.p. \\ \: \: is \: \: 3 \: \: times \\ \: \: the \: \: 17th \: \: term. \\ \\ \bold{solution : } \\ \tt the \: nth \: tem \: of \: an \: a.p \: is \: given \: by \\ \longmapsto \tt a_{n} = a_{1}(n - 1) \times d \\ \longmapsto \tt \: where \: a_{n} \: is \: the \: nth \: term \\ \tt a_{1} \: is \: the \: first \: term. \\ \\ \longmapsto\tt a_{1} \\ \longmapsto\tt a_{12} = 0 \\ a_{1} + 11d = 0 \\ \bold { a_{1} = - 11d \: \: \: (1)} \\ \\ \longmapsto \tt a_{n} \\ \longmapsto \tt \: a_{n} = 3 \times ( a_{17}) \\ \longmapsto \tt \: a_{n} = 3 \times ( a_{1} + 16d) \\ \longmapsto \tt \: a_{1} + (n - 1) \times d = 3 \times ( a_{1} + 16d) \\ \tt sebstitud \: the \: value \: of \: a_{1} \: from \\ \tt equation \: 1. \\ \\ \longmapsto \tt - 11d + (n - 1) \times d = 3( - 11d + 16d) \\ \longmapsto \tt \: - 11d + (n - 1) \times d = 3 \times 5d \\ \longmapsto \tt \: - 11d + (n - 1) \times d = 15d \\ \longmapsto \tt \: (n - 1) \times d = 26d \\ \longmapsto \tt \: n - 1 = 26d \div d \\ \longmapsto \tt \:n - 1 = 26 \\ \longmapsto \tt \: n = 26 + 1 \\ \longmapsto \tt \: n = 27 \\ \bold{i \: \: hope \: \: it \: \: help \: \: you}$