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If a(1)=3 and a(n)=n+a(n-1), the sum of ...

If `a_(1)=3 and a_(n)=n+a_(n-1)`, the sum of the first five term is

A

17

B

30

C

42

D

45

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we need to find the first five terms of the sequence defined by the recurrence relation \( a_n = n + a_{n-1} \) with the initial condition \( a_1 = 3 \). ### Step 1: Calculate \( a_2 \) Using the recurrence relation: \[ a_2 = 2 + a_{1} \] Substituting the value of \( a_1 \): \[ a_2 = 2 + 3 = 5 \] ### Step 2: Calculate \( a_3 \) Using the recurrence relation: \[ a_3 = 3 + a_{2} \] Substituting the value of \( a_2 \): \[ a_3 = 3 + 5 = 8 \] ### Step 3: Calculate \( a_4 \) Using the recurrence relation: \[ a_4 = 4 + a_{3} \] Substituting the value of \( a_3 \): \[ a_4 = 4 + 8 = 12 \] ### Step 4: Calculate \( a_5 \) Using the recurrence relation: \[ a_5 = 5 + a_{4} \] Substituting the value of \( a_4 \): \[ a_5 = 5 + 12 = 17 \] ### Step 5: Calculate the sum of the first five terms Now we need to find the sum: \[ S = a_1 + a_2 + a_3 + a_4 + a_5 \] Substituting the values we found: \[ S = 3 + 5 + 8 + 12 + 17 \] Calculating the sum: \[ S = 3 + 5 = 8 \] \[ S = 8 + 8 = 16 \] \[ S = 16 + 12 = 28 \] \[ S = 28 + 17 = 45 \] Thus, the sum of the first five terms is \( \boxed{45} \).
To solve the problem step by step, we need to find the first five terms of the sequence defined by the recurrence relation \( a_n = n + a_{n-1} \) with the initial condition \( a_1 = 3 \). ### Step 1: Calculate \( a_2 \) Using the recurrence relation: \[ a_2 = 2 + a_{1} \] Substituting the value of \( a_1 \): ...
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