Question

The first term of an AP is 5, the
last term is 45 and the sum of
all its terms is 40. find the
numbers of terms and the common difference of the AP .​

Answer 1

Correct Question :

The first term of an arithmetic progression is 5, the last term is 45 and the sum of all the terms is 400. find the number of terms and the common difference of the arithmetic progression.

Answer :

• The number of terms of the arithmetic progression = 16.
• The common difference of the arithmetic progression = 8 / 3.

Given :

• The first term (a) = 5.
• The last term (l) = 45.
• The sum of the terms (Sₙ) = 400.

To Find :

• The number of terms.
• The common difference of the arithmetic progression.

Solution :

Given,

The first term (a) = 5.

The last term (l) = 45.

The sum of (Sₙ) = 400.

Since, the last term of an A.P. is given.

So, we can use formula.

$\huge{ \green{ \star}} \: \: { \large{\boxed{ \blue{ s_{n} = \dfrac{n}{2}(a + l) }}}}$

We have to find the value of n.

• n = number of terms.

Now, substitute all the given values in the formula.

$: \implies \rm 400 = \dfrac{n}{2} (5 + 45) \\ \\ : \implies \rm 400 = \dfrac{n}{2} (50) \\ \\ : \implies \rm 400 = \dfrac{n}{2} \times 50 \\ \\ : \implies \rm400 = n \times 25 \\ \\ : \implies \rm \dfrac{400}{25} = n \\ \\ : \implies \rm 16 = n \\ \\ \: \: \rm \therefore \: \: n = 16$

Hence, the number of the terms is 16.

Now, we have to find the common difference of the arithmetic progression.

So, we can use this formula.

$\huge{ \green{ \star}} \: \: { \large{\boxed{ \blue{ s_{n} = \dfrac{n}{2}(2a + n - 1)d }}}}$

Where,

• d = common difference.

Now we have,

• a = 5.
• Sₙ = 400.
• n = 16.

$: \implies \rm 400 = \dfrac{16}{2} \bigg(2(5) + (16 - 1) d \bigg) \\ \\ : \implies \rm 400 = 8 (10 + 15d) \\ \\ : \implies \rm \dfrac{400}{8} = 10 + 15d \\ \\ : \implies \rm 50 = 10 + 15d \\ \\ : \implies \rm 50 - 10 = 15d \\ \\ : \implies \rm 40 = 15d \\ \\ : \implies \rm \dfrac{40}{15} = d \\ \\ : \implies \rm \dfrac{8}{3} = d \\ \\ \: \: \rm \therefore \: \: d = \dfrac{8}{3}$

Hence, the common difference (d) of the A.P. is $\sf \dfrac{8}{3}$

Hence,

The number of terms of an A.P. is 16 and the common difference (d) of an A.P. is $\sf \dfrac{8}{3}$