Question

The sum of first and seventh term of an A.P. is 6 and their product is 8. Find the first term and common difference.​

Answer 1

Question:-

➡ The sum of first and seventh term of an A.P. is 6 and their product is 8. Find the first term and common difference.

Solution:-

Let us assume that,

First Term = a and,

Common Difference = d

Now, we have to remember the formula to calculate the nth term of an A.P.

Formula to calculate nth term of an A.P. is

Nth term = a + (n - 1)d

So,

First term = a + (1 - 1)d = a ..... (i)

7th term = a + (7 - 1)d = a + 6d ....(ii)

Now, according to the given conditions,

➡ First term + 7th term = 6

➡ a + (a + 6d) = 6

➡ 2a + 6d = 6

➡ 2(a + 3d) = 6

Dividing both sides by 2, we get

➡ a + 3d = 3

➡ a = 3 - 3d

Similarly,

➡ 1st term × 7th term = 8

➡ a × (a + 6d) = 8

➡ (3 - 3d)(3 - 3d + 6d) = 8

➡ (3 - 3d)(3 + 3d) = 8

➡ 9 - 9d² = 8

➡ 9d² = 9 - 8

➡ 9d² = 1

➡ d² = 1/9

➡ d= ±1/3

So, when d = 1/3

First term (a)

= 3 - 3d

= 3 - 3 × 1/3

= 3 - 1

= 2

Again, when d = -1/3

First term (a)

= 3 - 3d

= 3 - 3 × -1/3

= 3 + 1

= 4.

➡ Hence, first term of the A.P. is either 2 or 4 and the common difference is either 1/3 or -1/3.

Verification:-

Let us verify our result.

When a = 2 and d = 1/3

a + (a + 6d)

= 2 + 2 + 6 × 1/3

= 4 + 2

= 6

Hence,

1st term + 7th term = 6

Again,

a(a + 6d)

= 2 × (2 + 6 × 1/3)

= 2 × (2 + 2)

= 2 × 4

= 8

Hence,

1st term × 7th term = 8

———————————————————————

Now, when a = 4 and d = -1/3

a + (a + 6d)

= 4 + (4 + 6 × -1/3)

= 4 + (4 + -2)

= 8 - 2

= 6

Hence,

1st term + 7th term = 6

Again,

a(a + 6d)

= 4 × (4 + 6 × -1/3)

= 4 × (4 - 2)

= 4 × 2

= 8

Hence,

1st term × 7th term = 8

Hence, the answer is correct. (Verified)

Answer:-

➡ Hence, first term of the A.P. is either 2 or 4 and the common difference is either 1/3 or -1/3.