9.the 5th term of an ap is 13, and the 15th term is 43. find the first term and common difference.\n10. find…

9.the 5th term of an ap is 13, and the 15th term is 43. find the first term and common difference.\n10. find the first four terms of the given sequence and tell whether the sequence is arithmetic or not.\na) ( t_{n}=5 - 3n ) b) ( t_{n}=2n + 1 )

9.the 5th term of an ap is 13, and the 15th term is 43. find the first term and common difference.\n10. find the first four terms of the given sequence and tell whether the sequence is arithmetic or not.\na) ( t_{n}=5 - 3n ) b) ( t_{n}=2n + 1 )

Answer

Explanation:

Step1: Use the formula for the nth term of an AP

The formula for the nth term of an arithmetic progression (AP) is (T_n=a+(n - 1)d), where (a) is the first term and (d) is the common difference. For the 5th term, (T_5=a+(5 - 1)d=a + 4d). Given (T_5 = 13), so (a+4d=13). For the 15th term, (T_{15}=a+(15 - 1)d=a+14d). Given (T_{15}=43), so (a + 14d=43).

Step2: Solve the system of equations

Subtract the first equation from the second equation: ((a + 14d)-(a + 4d)=43-13) (a+14d-a - 4d=30) (10d=30), so (d = 3).

Step3: Find the first term

Substitute (d = 3) into (a+4d=13). (a+4\times3=13) (a+12=13) (a=13 - 12=1).

Answer:

The first term (a = 1) and the common difference (d=3).