Question

for an A. P the first and last terms are 13 and 216 respectively. common difference is 7. find the sum of all terms.​

Answer 1

Answer :

• The sum of all the terms of the AP is 3435.

Explanation :

Given :

• First term of the AP, a = 13

• Last term of the AP, l = 216

• Common Difference of the AP, d = 7

To find :

• Sum of all the terms of the AP

Solution :

First we have to find the number of terms of the AP.

We know the formula for nth term of an AP i.e,

$\boxed{\bf{t_{n} = a_{1} + (n - 1)d}}$

Where :

• tn = nth term of the AP
• a1 = First term of the AP
• n = No. of terms of the AP
• d = Common difference

Now by using the formula for nth term of the AP and by substituting the values in it, we get :

$:\implies \bf{t_{n} = a_{1} + (n - 1)d}$

$:\implies \bf{216 = 13 + (n - 1)7}$

$:\implies \bf{216 - 13 = (n - 1)7}$

$:\implies \bf{203 = (n - 1)7}$

$:\implies \bf{203 = 7n - 7}$

$:\implies \bf{203 + 7 = 7n}$

$:\implies \bf{210 = 7n}$

$:\implies \bf{\dfrac{210}{7} = n}$

$:\implies \bf{30 = n}$

$\boxed{\therefore \bf{n = 30}}$

Hence the no. of terms of the AP is 30.

Now,

We know the formula for sum of no of terms of the AP i.e,

$\boxed{\bf{s_{n} = \dfrac{n}{2}\bigg(a_{1} + l\bigg)}}$

Where :

• sn = Sum of terms of the AP
• a1 = First term of the AP
• l = Last term of the AP

By using the formula for sum of terms of the AP

$:\implies \bf{s_{n} = \dfrac{n}{2}\bigg(a_{1} + l\bigg)}$

$:\implies \bf{s_{n} = \dfrac{30}{2}\bigg(13 + 216\bigg)}$

$:\implies \bf{s_{n} = 15 \times (13 + 216)}$

$:\implies \bf{s_{n} = 15 \times 229}$

$:\implies \bf{s_{n} = 3435}$

$\boxed{\therefore \bf{s_{n} = 3435}}$

Hence,

• The sum of all the terms of the AP is 3435.

Answer 2

Step-by-step explanation:

Given :

• Last term (A) = 216

• First term (a) = 13

• common difference (d) =7

To Find :

• find the sum of all terms.

Solution :

A = a + ( n - 1)d

Substitute all values :

216 = 13 + (n - 1)7

n - 1 = 203/7

n - 1 = 29

n = 29 + 1

n = 30

sum of all the terms S = n / 2 [ 2a + ( n - 1 ) d ]

Substitute all value :

= 30 / 2 [ 2 × 13 ( 30 - 1 ) × 7 ]

= 15 [ 26 + 203 ]

= 15 ( 229 )

= 3435

Hence the sum of all terms is 3435