Question

The 3rd term of an ap is 7 and it's 7th term is 2 more than three times the third term.find the sum of first 35 terms of the ap

Answer 1

Answer :

The sum upto 35 terms of the AP is 2345

Given :

• The 3rd term of an AP is 7
• The 7th term is 2 more than 3times the 3rd term

To Find :

• The sum of of first 35 terms of the AP

Formulae to be Used :

If a and d are first term and common difference of an AP nth term is given by :

$\bullet \: \: \:\bf a_n = a + (n - 1)d$

And the sum upto n terms is given by

$\bullet \: \: \bf S_n = \dfrac{n}{2} \{2a + (n - 1)d \}$

Solution :

Let us consider the first term be a and common difference be d

Now given ,

3rd term = 7

$\sf \implies a + (3 - 1)d = 7 \\ \\ \sf \implies a + 2d = 7 \: \: ..........(1)$

And again from the given statement :

7th = 3×(3rd term) + 2

$\implies \sf a + (7 - 1)d = 3 \{a + (3 - 1)d \} + 2 \\ \\ \implies \sf a + 6d = 3(a + 2d) + 2 \\ \\ \sf \implies a + 6d = 3a + 6d + 2 \\ \\ \sf \implies{a + 6d - 3a - 6d} = 2 \\ \\ \sf \implies - 2a = 2 \\ \\ \bf \implies a = - 1$

Using the value of a in (1) we have :

$\sf \implies{ - 1 + 2d = 7} \\ \\ \sf \implies{2d = 7 + 1} \\ \\ \sf \implies 2d = 8 \\ \\ \bf \implies d = 4$

Thus the first term is -1 and common difference is 4

Now sum upto 35 terms :

$\sf \implies S_{35} = \dfrac{35}{2} \{2 \times ( - 1) + (35 - 1)4 \} \\ \\ \sf \implies S_{35} = \dfrac{35}{2} \{ - 2 + 34 \times 4\} \\ \\ \sf \implies S_{35} = \dfrac{35}{2} \times 134 \\ \\ \sf \implies S_{35} = 35 \times 67 \\ \\ \sf \implies S_{35} = 2345$

Answer 2

GIVEN:

Third term of an AP = a3 = 7

The 7th term is 2 more then three times the 3rd term.

TO FIND:

The sum of first 35 terms of AP

SOLUTION:

Third term of AP = a3 = 7

Seventh term = a7 = 3(a3) + 2

= 3(7) + 2

= 21 + 2

= 23

Third term:

a + 2d = 7 -----(1)

Seventh term:

a + 6d = 23 -----(2)

Subtract both eq - (1) and (2)

a + 2d - a + 6d = 7 - 23

==> -4d = -16

==> d = 16/4

==> d = 4

Common Difference = 4

Substitute (d) in eq - (1) to find first term (a)

==> a + 2d = 7

==> a + 2(4) = 7

==> a + 8 = 7

==> a = 7 - 8

==> a = -1

First term = -1

We know that,

Sum of terms = n/2 [2a + (n - 1)d ]

= 35/2 [ 2(-1) + (35 - 1)4]

= 35/2 [ -2 + 136 ]

= 35/2 [ 134 ]

= 35 [ 67 ]

= 2345

Therefore, the sum of first 35 terms is 2345.