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The Sum of 4th and 8th Terms of an A.P. is 24 and the Sum of the 6th and 10th Terms is 34. Find the First Term and the Common Difference of the A.P. - CBSE Class 10 - Mathematics

Question

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Solution

In the given problem, the sum of 4th and 8th term is 24 and the sum of 6th and 10thterm is 34.

We can write this as,

a_4 + a_8 = 24  .....(1)

a_6 + a_10 = 34 ......(2)

We need to find a and d

For the given A.P., let us take the first term as a and the common difference as d

As we know,

a_n = a + (n - 1)d

For 4th term (n = 4),

a_4 = a + (4 -1)d

= a + 3d

For 8th term (n = 8)

a_8 = a + (8 - 1)d

= a + 7d

so on substituting the above values in 1 we get

(a + 3d) + (a + 7d) = 24

2a + 10d = 24.....(3)

Also for 6th term (n = 6)

a_6 = a + (6 - 1)d

= a + 5d

For 10 th term (n = 10)

a_10 = a + (10 - 1)d

= a + 9d

So on substituting the above values in 2 we get

(a +  5d)+(a +9d)= 34

2a + 14d = 34 ......(4)

Next we simplify 3 and 4. On substracting 3 from 4 we get

(2a + 14d) - (2a + 10d) = 34 - 24

2a + 14d - 3a - 10d = 10

4d =10

d = 10/4

d = 5/2

Further using the value of d in equation 3 we get

a + 10(5/2) = 24

2a + 5(5) = 24

2a = 24 - 25

On furthur simplifying we get

2a = -1

a = (-1)/2

Therefore for the given A.P a = (-1)/2 and d = 5/2

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Solution The Sum of 4th and 8th Terms of an A.P. is 24 and the Sum of the 6th and 10th Terms is 34. Find the First Term and the Common Difference of the A.P. Concept: nth Term of an AP.
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