2 + 10 + 18 +…………..+ n = 90. find number of terms in this series and also find the last term ‘n’

Problem Statement:

We are given the arithmetic series:

2 + 10 + 18 +…………..+ n = 90

We need to find:

1. The number of terms in this series.
2. The last term, (n).

Step 1: Understanding the Arithmetic Progression !

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference ((d)).

For the given series:

• The first term, a, is 2.
• The common difference, (d), is calculated as (10 – 2 = 8).

The sum of the first (n) terms of an arithmetic progression is given by the formula:

S_n = (n/2)(2a+(n-1)d)

Where:

• (Sn) is the sum of the first (n) terms.
• (n) is the number of terms.
• (a) is the first term.
• (d) is the common difference.

Step 2: Applying the Formula

We know that (Sn = 90), (a = 2), and (d = 8). Plugging these values into the formula gives us:

90 = (n/2)[2 (2) + (n – 1) 8]

Simplifying the expression inside the parentheses:

90 = (n/2)(4 + 8n – 8)

90 = (n/2) (8n – 4)

Multiply both sides by 2 to eliminate the fraction:

180 = n (8n – 4)

Distribute (n) on the right-hand side:

180 = 8n² – 4n

Rearrange the equation to form a quadratic equation:

8n² – 4n – 180 = 0

Divide the entire equation by 4 to simplify it:

2n² – n – 45 = 0

Step 3: Solving the Quadratic Equation

To solve this quadratic equation, we can use the factorization method. We need to find two numbers (p) and (q) such that:

pq = 2 x -45 = -90 ; p + q = -1

After considering the factors of (-90), we find:

p = -10 & q = 9

Now, split the middle term:

2n² – 10n + 9n – 45 = 0

Group the terms:

2n(n – 5) + 9(n – 5) = 0

Factor out the common terms:

(n – 5)(2n + 9) = 0

Set each factor to zero and solve for (n):

n – 5 = 0 OR 2n + 9 = 0

n = 5 OR n = -(9/2)

Since (n) must be a positive integer, we discard (n = -9/2) and accept (n = 5).

Thus, the series has 5 terms.

Step 4: Finding the Last Term

Now that we know there are 5 terms, let’s find the last term of the series, (an), using the formula for the nth term:

a_n = a + (n – 1) d

Substituting (a = 2), (d = 8), and (n = 5):

a₅ = 2 + (5 – 1) 8 = 2 + 32 = 34

Therefore, the last term of the series is 34.

Conclusion

In summary, the number of terms in the series is 5 and the last term is 34. This solution methodically shows how to apply the concepts of arithmetic progressions and quadratic equations to solve for unknowns in a sequence.