Question

in an ap given an=4d=2sn=14find n and a
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Answer 1

Given :-

• An = 4
• d = 2
• Sn = (-14).

To Find :-

• value of n and a ?

Formula used :-

In an AP with first term as a, common difference as d :-

• An = a + (n - 1)d
• Sn = (n/2)[2a + (n - 1)d]

Solution :-

Given that, An = 4 and d = 2 .

Putting both values in nth term formula of AP we get,

→ An = a + (n - 1)d

→ 4 = a + (n - 1)2

→ 4 = a + 2n - 2

→ 4 + 2 = a + 2n

→ a + 2n = 6

→ a = (6 - 2n) -------------- Eqn.(1)

Now, Putting given values of Sn = (-14), and d = 2 in sum formula of AP , we get,

→ (n/2)[2a + (n - 1)d] = Sn

→ (n/2)[2a+(n - 1)2]= (-14)

→ (n/2)[2a + 2n - 2] = (-14)

→ (n/2) * 2 * [ a + n - 1] = (-14)

→ n * [ a + n - 1 ] = (-14)

→ an + n² - n = (-14)

Putting value of a from Eqn.(1) Now,

→ (6 - 2n)n + n² - n= (-14)

→ 6n - 2n² + n² - n = (-14)

→ -n² + 5n = (-14)

Taking (-1) common from both sides now,

→ (-1) * (n² - 5n) = (-1) * 14

→ n² - 5n = 14

→ n² - 5n - 14 = 0

→ n² - 7n + 2n - 14 = 0

→ n(n - 7) + 2(n - 7) = 0

→ (n - 7)(n + 2) = 0

Putting both equal to zero now, we get,

→ n - 7 = 0

→ n = 7 .

Or,

→ n + 2 = 0

→ n = (-2)

Since value of n can't be Negative.

Therefore ,

→ n = 7. (Ans.)

Putting value of n in Eqn.(1) Now,

→ a = 6 - 2n

→ a = 6 - 2*7

→ a = 6 - 14

→ a = (-8) (Ans.)

Hence, First term (a) of given AP is (-8) and Total terms(n) is 7.