To analyze the given assertion (A) and reason (R) in the context of the problem, we need to confirm the validity of each statement and see if the reason appropriately explains the assertion.
### Given:
- [tex]$a_n = 9 - 5n$[/tex]
- [tex]$n = 6$[/tex]
### Assertion (A):
"The sum of the series with the nth term [tex]\(a_n = 9 - 5n\)[/tex] is 220 when the number of terms [tex]\(n = 6\)[/tex]."
### Reason (R):
"The sum of the first [tex]\(n\)[/tex] terms in an arithmetic progression (A.P.) is given by the formula [tex]\( S_n = \frac{n}{2} [2a + (n-1)d] \)[/tex], where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference."
### Steps:
1. Identify the first term ([tex]\(a\)[/tex]) and common difference ([tex]\(d\)[/tex]) of the series:
- The first term ([tex]\(a\)[/tex]) when [tex]\(n=1\)[/tex]:
[tex]\[
a_1 = 9 - 5 \cdot 1 = 4
\][/tex]
- The second term ([tex]\(a_2\)[/tex]) when [tex]\(n=2\)[/tex]:
[tex]\[
a_2 = 9 - 5 \cdot 2 = -1
\][/tex]
- Compute the common difference [tex]\(d\)[/tex]:
[tex]\[
d = a_2 - a_1 = -1 - 4 = -5
\][/tex]
2. Calculate the sum (S6) using the sum formula for an arithmetic progression:
- Using the formula [tex]\( S_n = \frac{n}{2} [2a + (n-1)d] \)[/tex]:
[tex]\[
S_6 = \frac{6}{2} [2 \cdot 4 + (6-1) \cdot (-5)]
\][/tex]
[tex]\[
S_6 = 3 [2 \cdot 4 + 5 \cdot (-5)]
\][/tex]
[tex]\[
S_6 = 3 [8 - 25]
\][/tex]
[tex]\[
S_6 = 3 \times (-17)
\][/tex]
[tex]\[
S_6 = -51
\][/tex]
3. Compare the calculated sum (S6) with the given assertion:
- The assertion claims that the sum is 220, but the calculated sum using the arithmetic progression formula is -51.
Since our calculated sum does not match the assertion in (A), it means that Assertion (A) is false.
4. Evaluate the reason (R):
- The formula provided for the sum of the first [tex]\(n\)[/tex] terms in an arithmetic progression is correct and is a standard mathematical formula.
### Conclusion:
1. Assertion (A) is false because our calculation indicates that the sum of the series is not 220.
2. Reason (R) is true as it correctly states the formula for the sum of an arithmetic progression.
Therefore, the correct choice is:
(d) Assertion (A) is false but reason (R) is true.
