Present value of annuity with intermitent geometric and arithmetic gradient

[DexterOnline]

Help me find a closed form formula that determines the present value of an annuity that has intermittent gradients. The gradients switch with each term between geometric and arithmetic

The geometric gradient is 10%
The arithmetic gradient is $10

The discount rate is 12% compounded monthly

T	PMT	Growth	Discount	PV
1	100	Geometric	(1.12)^(-1)	89.29
2	100	Arithmetic	(1.12)^(-2)	79.72
3	110	Geometric	(1.12)^(-3)	78.3
4	110	Arithmetic	(1.12)^(-4)	69.91
5	121	Geometric	(1.12)^(-5)	68.66
6	120	Arithmetic	(1.12)^(-6)	60.8
7	133.1	Geometric	(1.12)^(-7)	60.21
8	130	Arithmetic	(1.12)^(-8)	52.5
9	146.41	Geometric	(1.12)^(-9)	52.8
10	140	Arithmetic	(1.12)^(-10)	45.08
11	161.051	Geometric	(1.12)^(-11)	46.3
12	150	Arithmetic	(1.12)^(-12)	38.5
			NPV	742.07

 

[DexterOnline]

Just before I say Good Bye Mr. Chips, I would like to make the problem bit more fun by allowing intermittent types of interest that switch back and forth into compound and then simple interest

The question still stands to find a closed form formula that determines the present value of such an annuity.

T	PMT	Growth	Interest type	Discount	PV
1	100	Geometric	Compound	(1.12)^(-1)	89.29
2	100	Arithmetic	Compound	(1.12)^(-2)	79.72
3	110	Geometric	Simple	(1+0.12x3)^(-1)	78.3
4	110	Arithmetic	Simple	(1+0.12x4)^(-1)	69.91
5	121	Geometric	Compound	(1.12)^(-5)	68.66
6	120	Arithmetic	Compound	(1.12)^(-6)	60.8
7	133.1	Geometric	Simple	(1+0.12x7)^(-1)	60.21
8	130	Arithmetic	Simple	(1+0.12x8)^(-1)	52.5
9	146.41	Geometric	Compound	(1.12)^(-9)	52.8
10	140	Arithmetic	Compound	(1.12)^(-10)	45.08
11	161.051	Geometric	Simple	(1+0.12x11)^(-1)	46.3
12	150	Arithmetic	Simple	(1+0.12x12)^(-1)	38.5

 

[Ishuda]

Help me find a closed form formula that determines the present value of an annuity that has intermittent gradients. The gradients switch with each term between geometric and arithmetic

The geometric gradient is 10%
The arithmetic gradient is $10

The discount rate is 12% compounded monthly

T	PMT	Growth	Discount	PV
1	100	Geometric	(1.12)^(-1)	89.29
2	100	Arithmetic	(1.12)^(-2)	79.72
3	110	Geometric	(1.12)^(-3)	78.3
4	110	Arithmetic	(1.12)^(-4)	69.91
5	121	Geometric	(1.12)^(-5)	68.66
6	120	Arithmetic	(1.12)^(-6)	60.8
7	133.1	Geometric	(1.12)^(-7)	60.21
8	130	Arithmetic	(1.12)^(-8)	52.5
9	146.41	Geometric	(1.12)^(-9)	52.8
10	140	Arithmetic	(1.12)^(-10)	45.08
11	161.051	Geometric	(1.12)^(-11)	46.3
12	150	Arithmetic	(1.12)^(-12)	38.5
			NPV	742.07

To start with, break it into two pieces. You then have two series which you should be able to easily get the closed form solution.